The best directorial debuts

trying to leave aside what's been named already...

repo man
bottle rocket (not an anderson obsessive myself, but i mostly enjoy his films, & i think this is pretty great).
the evil dead
l'age d'or
gates of heaven
eraserhead (i actually don't really love eraserhead, but i like almost everything else he's done, & it's certainly revered by lots of people).
the texas chainsaw massacre
performance

Nil by Mouth (1997) - Gary Oldman. ⤵
https://m.imdb.com/title/tt0119792/?ref=m_nv_sr_1

I gotta give some love to Jordan Peele for Get Out...one of my favorite films of 2017. The fact that he was known almost entirely for comedy prior to this makes it even more impressive.

Not mentioned so far:

They Live By Night - Ray
Night of the Hunter - Laughton
400 Blows -Truffaut
Breathless - Godard
Who's Afraid of Virginia Woolf? - Nichols
Badlands - Malick
The Spirit of the Beehive - Erice
The Seventh Continent - M. Haneke
Welcome to the Dollhouse - Solondz
Gattaca - Niccol
Synecdoche NY - Kaufman
Son of Saul - Nemes

Few film-makers ever make a film as good as any of these debuts!

Pi - Darren Aronofsky

Steven Soderbergh; Sex, Lies, and Videotape
Michael Moore; Roger and Me
both from 1989

The Boondock Saints- Troy Duffy

Eraserhead