Understanding Kh1 and Be3 in the King's Indian Defense Bayonet Attack In this position (diagram), there have been two games where White played 14.Kh1 followed by 15.Be3, then Qe1 and Bf2.  


What is the idea behind 14.Kh1 and the other kingside moves when Black is not developing anything strong on the kingside? 


The opening was the King's Indian Defense Bayonet Attack.  Black himself had played 13...Ne7-g8, with the Knight apparently headed for f6 and possibly h5.  Clearly Black is playing on the kingside, and White has ideas of a2-a4 and cxd6 on the queenside. 

The position in FEN is


  r1bq1rnk/ppp3bp/3p2p1/2PPpp2/1PN1Pn2/2N2P2/P3B1PP/R1BQ1RK1 w - - 0 14


for those who wish to paste it and make a search of the 6 games with this position. By the way, Stockfish 11 (at depth 49) also evaluates favorably towards Kh1 and Be3:

 [title "First line, with Stockfish 11 (depth 49) evaluation = 0.95"]
 [fen "r1bq1rnk/ppp3bp/3p2p1/2PPpp2/1PN1Pn2/2N2P2/P3B1PP/R1BQ1RK1 w - - 0 14"]

 1. Kh1 fxe4 2. fxe4 Nf6 3. Be3 b6 4.c6 h5 5.a4 Ng4 6.Bg1 a6 7.Bf3 Kg8 8.Rb1 Bh6 9.Qc2 Qe7 10.b5 axb5 11.Nxb5 Ba6 12.Rb3 Nf6 13.Rfb1 Rab8 14.Ra1 h4 15.Ne3 N6h5 16.Bf2 Bc8 17.Nc4 Ba6 18.Be3 Ng3+ 19.Kg1 Ngh5 20.Bxh5 gxh5 21.Kh1 Qg7 22.h3 Ra8 23.Rbb1 Bxb5 24.axb5 Rxa1 25.Rxa1 Qg3 26.Bg1 Qd3 27.Qxd3 


And Be3:

 [title "Second line, with Stockfish 11 (depth 49) evaluation = 0.68"]
 [fen "r1bq1rnk/ppp3bp/3p2p1/2PPpp2/1PN1Pn2/2N2P2/P3B1PP/R1BQ1RK1 w - - 0 14"]

 1.Be3 Qg5 2.g3 Nxe2+ 3.Qxe2 f4 4.Bd2 Qe7 5.cxd6 cxd6 6.a4 Nh6 7.Nd1 Nf7 8.Nf2 Ng5 9.Ra3 Bd7 10.Kg2 Rac8 11.b5 h5 12.Na5 Rc2 13.Rc1 Rxc1 14.Bxc1 Bg4 15.Bd2 Kg8 16.Nc4 Bd7 17.Rc3 Rc8 18.Qd1 h4 19.g4 Be8 20.Rc2 a6 21.Bb4 axb5 22.Nxd6 Rxc2 23.Qxc2 bxa4 24.Qc8 Bf8 25.Nc4 Qxb4 26.Qxe8 Qxc4 27.Qxg6+ Kh8 28.Qxg5 


Update:

After depth 50/83, Stockfish 11 likes 14.Kh1 second best and calculates that White can stand up to any attack on the kingside and so chooses to proceed with 14.a4 followed by cxd6, though it still likes Be3:


+/- (0.97): 14.a4 Nf6 15.Be3 g5 16.cxd6 cxd6 17.Rc1 Qe7 18.Re1 fxe4 19.Nxe4 N6xd5 20.Bxf4 Nxf4 21.Qxd6 Qe8 22.Ne3 g4 23.fxg4 Be6 24.Rcd1 Rd8 25.Qxd8 Qxd8 26.Rxd8 Rxd8 27.g3 Nxe2+ 28.Rxe2 Rd4 29.Nc5 Bc8 30.Rc2 b6 31.Nb3 Rd8 32.Nd2 Bh6 33.Ndc4 Be6 34.Kf2 Rc8 35.Kf3 Bxe3 36.Nxe3 Rxc2 37.Nxc2 Bb3 38.Ne3 Bxa4 39.Ke4 Bc6+ 40.Kxe5 a5 41.bxa5 bxa5 42.Kd6 Be4 43.Nc4 Kg7 44.Kc5 Kf6 
+/- (0.78): 14.Kh1 h5 15.Bd2 fxe4 16.fxe4 Nf6 17.a4 Ng4 18.h3 Nh6 19.Qe1 Bd7 20.Rg1 Nf7 21.Be3 g5 22.Bf1 h4 23.Be2 a6 24.Rf1 Rb8 25.Rb1 Qf6 26.b5 dxc5 27.Bxc5 Rfc8 28.Ne3 Bf8 29.Bxf8 Rxf8 30.Bg4 Bxg4 31.Nxg4 Qe7 32.bxa6 bxa6 33.Qf2 Nd6 34.Qa7 Ra8 35.Qc5 Qg7 36.Qc6 Nh5 37.Rxf8+ Rxf8 38.Qxa6 Ng3+ 39.Kh2 Ndxe4 40.Nxe4 Nxe4 


I will test out the novelty 14.a4 (never played in this position before) and see if Stockfish 11 playing White wins against itself with depth 50+ for each move.  I am curious to see how strong Black's kingside attack really is, keeping in mind the answers given so far that suggest there can be a strong attack there for Black.
I found the games, and you have to take them with a grain of salt since one was between a 2150, and someone who was not even FIDE rated. The second was between two 2360's, which I find a bit suspicious. I am going to post that second game.

I can tell you the basic ideas behind each move, but I am not sure that they were strung together in an optimal manner. This whole line is pretty rare, and as of Ng8, there is only one GM game in Mega2020.

The idea behind:


Ng8 - The knight is not well positioned on e7, so it is going to g8-f6-h5 to join in the attack. In addition, it will over protect the Bg7 should it go to h6 in an attempt to trade off the bad bishop.
Kh1 - To make room for the Be3-g1 maneuver, which did not happen in these games, in an attempt to avoid the potential trade of the Be3 for the Bg7 (after Bh6).
In both cases, in the game below, the Qe1 and Bf2 were to overprotect the g3 square since a Nh5 often sacrifices itself there, and it can be incredibly dangerous. Based on this common idea, I do not think that Kh1 fit in with this position.


Here is one of the games with the maneuvers you mentioned.

 [Event "Paks Atom Cup GM 1st"]
 [Site "Paks"]
 [Date "1996.??.??"]
 [Round "?"]
 [White "Vukanovic, Sasa"]
 [Black "Molnar, Bela"]
 [Result "0-1"]
 [ECO "E97"]
 [WhiteElo "2360"]
 [BlackElo "2360"]
 [PlyCount "108"]
 [EventDate "1996.09.??"]
 [EventType "tourn"]
 [EventRounds "13"]
 [EventCountry "HUN"]
 [EventCategory "7"]
 [FEN ""]

 1. Nf3 g6 2. c4 Bg7 3. e4 d6 4. d4 Nf6 5. Nc3 O-O 6. Be2 e5 7. O-O Nc6 8. d5 Ne7 9. b4 Nh5 10. c5 Nf4 11. Nd2 f5 12. f3 Kh8 13. Nc4 Ng8 14. Kh1 Nf6 15. Be3 N6h5 16. Qe1 Nxe2 17. Qxe2 f4 18. Bf2 Rg8 19. cxd6 cxd6 20. Nb5 Bf8 21. Rfc1 b6 22. a4 Ba6 23. Nc3 g5 24. b5 Bc8 25. a5 Rb8 26. axb6 axb6 27. Na4 g4 28. Ncxb6 g3 29. Rxc8 Rxc8 30. Nxc8 gxf2 31. Qxf2 Qxc8 32. b6 Be7 33. Qb2 Qd7 34. b7 Rb8 35. Nc5 Qc7 36. Ne6 Qc4 37. Kg1 Nf6 38. Rc1 Rxb7 39. Rxc4 Rxb2 40. Rc7 Ng8 41. g3 fxg3 42. hxg3 h6 43. Rd7 Kh7 44. f4 Kg6 45. fxe5 dxe5 46. d6 Bf6 47. Rc7 Rb6 48. Rd7 Kh5 49. Nc7 Rb7 50. Kf2 Kg6 51. Kf3 h5 52. Ke3 Nh6 53. Kd3 Nf7 54. Kc4 Nxd6+ 0-1


The only GM game from 13...Ng8, so you can see how they played it.

 [Event "Gashimov Memorial 3rd"]
 [Site "Shamkir"]
 [Date "2016.05.28"]
 [Round "3"]
 [White "Safarli, Eltaj"]
 [Black "Mamedov, Rauf"]
 [Result "1/2-1/2"]
 [ECO "E97"]
 [WhiteElo "2664"]
 [BlackElo "2655"]
 [PlyCount "55"]
 [EventDate "2016.05.26"]
 [EventType "tourn"]
 [EventRounds "9"]
 [EventCountry "AZE"]
 [EventCategory "20"]
 [FEN ""]

 1. d4 Nf6 2. c4 g6 3. Nc3 Bg7 4. e4 d6 5. Nf3 O-O 6. Be2 e5 7. O-O Nc6 8. d5 Ne7 9. b4 Kh8 10. c5 Nh5 11. Nd2 Nf4 12. Nc4 f5 13. f3 Ng8 14. cxd6 cxd6 15. Be3 Bh6 16. a4 Bd7 17. Nb5 Bxb5 18. axb5 fxe4 19. fxe4 Nxe2+ 20. Qxe2 Rxf1+ 21. Rxf1 Bxe3+ 22. Qxe3 Nf6 23. Qh3 Nxe4 24. Qg4 Nf6 25. Qe6 Ne4 26. Qg4 Nf6 27. Qe6 Ne4 28. Qg4 1/2-1/2