If White doesn't play c4 then you're in a Pirc.
Any 1.e4 defence similar to the King's Indian Defence?
"Building a repertoire ... we will take the idealized situation of someone starting from square one ... If you choose the Pirc against 1 e4, it makes sense to consider the King's Indian against 1 d4. ..." - GM John Nunn (1998)

No there isn't... In the Pirc/modern Black rarely if ever plays on the kingside. The only way I think you can get regular attacks on the kingside is to try the so-called "black lion" defense but it's not so easy to use:

You can reduce your opening preparation by learning the Pirc and the King's Indian together (1. ... d6 agasint any White move) , but the Pirc is not going to give you the fun attacks that you get in the King's Indian, with a big pawn roller charging at White's king, culminating in a Bishop sac at h3

Pirc Modern has nothing to do with the King's Indian, there are no closed positions and you are the one getting MATED.
Look for closed openings, like the Ruy Lopez g6 setups or the French.

The Black Lion seems pretty toothless (excuse the pun) and passive in the main line. You also get these early g4 and f4 lines which are quite dangerous.
In the Pirc, as others have said, Black generally attacks on the queenside whereas White attacks on the kingside. Same with the Dragon.
The Ruy Lopez lines with ...g6 might be your best bet. Other than that, there's the Sicilian Sveshnikov where Black often goes for ...f5 breaks, but it's a more positionally dominated game. If you're not worried about your opening being entirely sound, there's Owen's Defence (1...b6), which is a bit offbeat, but can lead to White closing the centre. Something like this:

At us mortals' level, I agree. White shouldn't close the centre, though. 6.exd5 Nxd5 7.O-O is pleasant for White. I'm always going to dislike it as I used it in the decisive final round of a blitz tournament (which I wasn't taking entirely seriously), lost the game and, as I later found out, £490(!)

If you are strong at understanding the King's Indian Defense, it's not about attacking the Kingside that you understand, it's understanding how to execute "blocked positions", and the "Pawn-Pointing Theory".
In the Classical King's Indian, the center isn't just mildly closed like say, the QGD, it's outright blocked by pawns. White pawns on f3, e4, d5. Black pawns on d6, e5, f4. Observe which direction these pawns are pointing. White's pawns point toward the Queenside, Black's the Kingside, and hence that is the direction that they attack.
Against 1.e4, the opening that most frequently leads to a blocked center is the French Defense, and is what I always recommend to advocates of the King's Indian Defense.
However, do note! In the French, the pawns point the opposite way. That said, doesn't matter if White castles Queenside or Kingside, Black's attack is on the Queenside. Not all attacks are direct attacks on the King. Some could be trainwrecking the other side of the board and forcing either a promotion or the win of material via the opponent having to surrender material to avoid the promotion. It could also mean simply surviving the immediate attack on your King and winning the endgame because you opponent's pawns are a trainwreck mess.
So again, I'm going to say the same thing I've said in numerous other posts. If you advocate the King's Indian, play the French. If you advocate the French, play the King's Indian!

The Owen's is completely sound, just not popular. Lol.
It is sound... It's just a really cramped position for Black provided he doesn't just blunder away.

The Black Lion is just a cool way to market the Hanham Philidor lol.
No the Hanham is better. If you play 1.e4 d6 2.d4 Nf6 3.Nc3 Nbd7 you have to deal with 4.f4. The Hanham avoids that.

The Owen's is completely sound, just not popular. Lol.
Prove with analysis the Owen's is not sound or GTFO.

The Black Lion is just a cool way to market the Hanham Philidor lol.
No the Hanham is better. If you play 1.e4 d6 2.d4 Nf6 3.Nc3 Nbd7 you have to deal with 4.f4. The Hanham avoids that.
Which is why the Lion is best entered via 1.e4 d6 2.d4 Nf6 3.Nc3 e5 4.Nf3 Nbd7. But even then there's 5.g4!? which is good fun for White.

The Black Lion seems pretty toothless (excuse the pun) and passive in the main line. You also get these early g4 and f4 lines which are quite dangerous.
In the Pirc, as others have said, Black generally attacks on the queenside whereas White attacks on the kingside. Same with the Dragon.
The Ruy Lopez lines with ...g6 might be your best bet. Other than that, there's the Sicilian Sveshnikov where Black often goes for ...f5 breaks, but it's a more positionally dominated game. If you're not worried about your opening being entirely sound, there's Owen's Defence (1...b6), which is a bit offbeat, but can lead to White closing the centre. Something like this:
And now white uses his a-pawn as a battering ram.
Which is why Black plays ...a6 at some point (to allow him to reply to a5 with ...b5). It's not quite that simple. Meanwhile Black's playing stuff like ...f6 and ...g5 etc. with an attack on the opposite wing with suitable preparation.
Since I seem to score pretty high with the KIA against openings such as the French (they get thrown off-guard and don't generate any queenside counterplay) and allow me to roll my pawns and pieces at their castled king), I think I might be quite good at a defence which allows me to lock the center early on and causes Black to attack kingside (and White queenside).
Besides from a transposition to the KID from something like the Modern Defence:
And the Sicilian Dragon (which isn't really that similar to the KID, since it only allows you to attack queenside):