Specifically, I would try to see if there are any (probable) primes with 1000000 or more, but fewer than 1000100, digits. There are 3954322 Leyland numbers in this range but by sieving out ones that are divisible by small primes — say, up to 10^11 — only about 61000 should remain. The sieving can be done in three weeks and only then would I start the search. Each remaining candidate needs about five hours to decide if it was composite, which comes to 35 years overall but, fortunately, I can distribute this across 72 Mac-mini cores, so six months. I will likely add some cores to the job but still, five months!
I have created a dictionary of Leyland (x,y) pairs from (999999,10) to (1000999,10), sorted by magnitude and preceded by its Leyland-number index (21588818851 to 21628375832). The text file is more than a gigabyte so I don't see much utility in linking to it. Here is a much abridged version showing the initial-, middle-, and final-100 entries:
21588818851 (999999,10)
21588818852 (333333,1000)
21588818853 (227361,25020)
21588818854 (191945,162114)
21588818855 (307560,1784)
21588818856 (193629,146053)
21588818857 (196205,124941)
21588818858 (192901,152757)
21588818859 (194076,142107)
21588818860 (200954,94680)
21588818861 (201859,89939)
21588818862 (192251,159048)
21588818863 (218510,37709)
21588818864 (195348,131542)
21588818865 (220103,34940)
21588818866 (191577,165893)
21588818867 (190250,180400)
21588818868 (208345,63056)
21588818869 (232052,20388)
21588818870 (200835,95325)
21588818871 (191764,163960)
21588818872 (207423,66231)
21588818873 (197894,113033)
21588818874 (206956,67911)
21588818875 (189903,184434)
21588818876 (193737,145088)
21588818877 (201347,92586)
21588818878 (217470,39658)
21588818879 (236785,16720)
21588818880 (197290,117132)
21588818881 (226287,26252)
21588818882 (196255,124568)
21588818883 (194241,140682)
21588818884 (189848,185083)
21588818885 (191070,171269)
21588818886 (207015,67696)
21588818887 (203854,80439)
21588818888 (307884,1770)
21588818889 (209900,58099)
21588818890 (208241,63405)
21588818891 (192010,161457)
21588818892 (194387,139435)
21588818893 (205547,73293)
21588818894 (191075,171215)
21588818895 (202991,84396)
21588818896 (192289,158672)
21588818897 (232494,20007)
21588818898 (192986,151956)
21588818899 (200313,98216)
21588818900 (202944,84618)
21588818901 (237012,16565)
21588818902 (214086,46883)
21588818903 (200103,99408)
21588818904 (203511,81985)
21588818905 (194104,141864)
21588818906 (211140,54474)
21588818907 (191597,165685)
21588818908 (214213,46585)
21588818909 (197582,115129)
21588818910 (202966,84514)
21588818911 (322185,1270)
21588818912 (194740,136473)
21588818913 (317076,1425)
21588818914 (200702,96052)
21588818915 (192426,157325)
21588818916 (195063,133827)
21588818917 (190278,180079)
21588818918 (199423,103386)
21588818919 (192875,153003)
21588818920 (196264,124501)
21588818921 (212678,50343)
21588818922 (215727,43200)
21588818923 (233349,19294)
21588818924 (194183,141181)
21588818925 (192774,153963)
21588818926 (224563,28385)
21588818927 (220403,34446)
21588818928 (217691,39234)
21588818929 (194955,134705)
21588818930 (219881,35311)
21588818931 (195163,133020)
21588818932 (237971,15929)
21588818933 (200461,97386)
21588818934 (200949,94707)
21588818935 (201451,92041)
21588818936 (194377,139520)
21588818937 (190731,174977)
21588818938 (192678,154882)
21588818939 (212303,51315)
21588818940 (245582,11802)
21588818941 (193095,150936)
21588818942 (197073,118647)
21588818943 (189738,186389)
21588818944 (189550,188646)
21588818945 (190635,176044)
21588818946 (190134,181737)
21588818947 (245425,11873)
21588818948 (192078,160773)
21588818949 (214120,46803)
21588818950 (195110,133447)
...
21608597292 (202096,89251)
21608597293 (213266,49127)
21608597294 (210544,56490)
21608597295 (195707,129488)
21608597296 (195000,135134)
21608597297 (222742,31027)
21608597298 (197714,114904)
21608597299 (193636,146861)
21608597300 (194915,135832)
21608597301 (195622,130152)
21608597302 (196402,124205)
21608597303 (206332,70628)
21608597304 (201718,91178)
21608597305 (191155,171383)
21608597306 (222218,31793)
21608597307 (191257,170285)
21608597308 (199239,105100)
21608597309 (222418,31498)
21608597310 (212447,51216)
21608597311 (194032,143338)
21608597312 (203365,83123)
21608597313 (214595,45948)
21608597314 (216053,42737)
21608597315 (234319,18613)
21608597316 (197760,114593)
21608597317 (221076,33542)
21608597318 (231213,21241)
21608597319 (218728,37512)
21608597320 (203171,84027)
21608597321 (207838,65139)
21608597322 (230038,22350)
21608597323 (192755,155068)
21608597324 (333020,1010)
21608597325 (199691,102385)
21608597326 (212181,51917)
21608597327 (230616,21796)
21608597328 (193347,149496)
21608597329 (213301,49040)
21608597330 (192266,159854)
21608597331 (221682,32600)
21608597332 (231026,21413)
21608597333 (239495,15050)
21608597334 (230542,21866)
21608597335 (194521,139123)
21608597336 (211086,54925)
21608597337 (195628,130105)
21608597338 (193279,150124)
21608597339 (251796,9407)
21608597340 (202881,85400)
21608597341 (214795,45491)
21608597342 (190804,175225)
21608597343 (248669,10554)
21608597344 (193007,152667)
21608597345 (193242,150467)
21608597346 (235964,17380)
21608597347 (199666,102533)
21608597348 (228061,24377)
21608597349 (499171,101)
21608597350 (190229,181738)
21608597351 (199984,100669)
21608597352 (235914,17416)
21608597353 (223578,29850)
21608597354 (204486,78118)
21608597355 (221293,33201)
21608597356 (216472,41864)
21608597357 (197252,118083)
21608597358 (192998,152752)
21608597359 (200013,100501)
21608597360 (297398,2313)
21608597361 (208174,63984)
21608597362 (196614,122644)
21608597363 (194347,140606)
21608597364 (194583,138599)
21608597365 (199099,105958)
21608597366 (309709,1700)
21608597367 (211028,55090)
21608597368 (193434,148697)
21608597369 (267212,5549)
21608597370 (362050,580)
21608597371 (190527,178328)
21608597372 (196705,121981)
21608597373 (219270,36548)
21608597374 (193884,144643)
21608597375 (191521,167481)
21608597376 (225704,27089)
21608597377 (199057,106217)
21608597378 (201016,94888)
21608597379 (212300,51602)
21608597380 (202625,86634)
21608597381 (198417,110257)
21608597382 (249059,10402)
21608597383 (207699,65624)
21608597384 (196679,122170)
21608597385 (218412,38088)
21608597386 (192388,158644)
21608597387 (209323,60213)
21608597388 (395825,337)
21608597389 (194211,141778)
21608597390 (209893,58440)
21608597391 (191644,166193)
... 21628375733 (194738,138113)
21628375734 (208965,61698)
21628375735 (202992,85354)
21628375736 (216089,42889)
21628375737 (194756,137962)
21628375738 (195035,135646)
21628375739 (346049,781)
21628375740 (213162,49654)
21628375741 (255139,8382)
21628375742 (214868,45569)
21628375743 (247864,10927)
21628375744 (191386,169927)
21628375745 (206738,69482)
21628375746 (191987,163640)
21628375747 (208753,62393)
21628375748 (198003,113625)
21628375749 (195603,131070)
21628375750 (195340,133166)
21628375751 (203582,82591)
21628375752 (196155,126795)
21628375753 (228293,24250)
21628375754 (207358,67204)
21628375755 (192103,162458)
21628375756 (195921,128587)
21628375757 (271246,4902)
21628375758 (190526,179420)
21628375759 (192536,158133)
21628375760 (212552,51219)
21628375761 (204456,78689)
21628375762 (220723,34284)
21628375763 (214264,46968)
21628375764 (216663,41694)
21628375765 (217445,40129)
21628375766 (198420,110879)
21628375767 (244895,12231)
21628375768 (195726,130103)
21628375769 (202620,87152)
21628375770 (192800,155562)
21628375771 (189723,188846)
21628375772 (205892,72739)
21628375773 (204280,79457)
21628375774 (218032,39000)
21628375775 (220514,34625)
21628375776 (208691,62598)
21628375777 (190277,182283)
21628375778 (215694,43735)
21628375779 (190181,183401)
21628375780 (191137,172614)
21628375781 (193893,145424)
21628375782 (209580,59733)
21628375783 (192689,156637)
21628375784 (203883,81222)
21628375785 (195515,131767)
21628375786 (200024,101017)
21628375787 (199138,106331)
21628375788 (221281,33393)
21628375789 (194968,136198)
21628375790 (200965,95711)
21628375791 (212198,52154)
21628375792 (197398,117752)
21628375793 (190287,182167)
21628375794 (207975,65024)
21628375795 (192049,163007)
21628375796 (214901,45494)
21628375797 (236652,16976)
21628375798 (221066,33733)
21628375799 (201434,93189)
21628375800 (201794,91306)
21628375801 (242759,13287)
21628375802 (204227,79690)
21628375803 (203296,83917)
21628375804 (201754,91513)
21628375805 (190106,184280)
21628375806 (204800,77213)
21628375807 (199269,105525)
21628375808 (192448,159001)
21628375809 (202927,85665)
21628375810 (212929,50245)
21628375811 (209859,58866)
21628375812 (195899,128757)
21628375813 (190940,174775)
21628375814 (205840,72945)
21628375815 (200294,99460)
21628375816 (198080,113112)
21628375817 (286501,3118)
21628375818 (257686,7666)
21628375819 (286892,3084)
21628375820 (224017,29404)
21628375821 (195026,135720)
21628375822 (197542,116754)
21628375823 (193314,150695)
21628375824 (191768,165899)
21628375825 (194701,138424)
21628375826 (197442,117446)
21628375827 (197971,113839)
21628375828 (192497,158517)
21628375829 (208583,62957)
21628375830 (223309,30379)
21628375831 (238055,16029)
21628375832 (1000999,10)