Is "Racketeer Rabbit" (1946).
I don't know why this one isn't very famous; I guess it's because Friz Freleng covered some of the same ground later with "Bugs and Thugs" (1954), which became one of the best-known Bugs cartoons ("You might, rabbit, you might" is one of the most-quoted lines from any cartoon). "Bugs and Thugs" is good, but "Racketeer" is just about perfect. Even by the standards of 1946, almost certainly the best year for Warner Brothers cartoon releases -- Clampett (in his last cartoons for the studio) Jones and Freleng all in peak form, the first cartoons from McKimson and Davis -- I've always thought this one was special.
The gags are from Mike Maltese and Friz Freleng's top drawer: the "and me, boss?" sequence, the "making something really fast just so you can hit the villain with it" scene, the "hiding Rocky" scene, and of course the amazingly written and timed scene where Bugs actually gets into costume to impersonate a policeman (even though Rocky can't see him doing it and there's no reason for him to do it). It has several of my favorite Maltese lines of dialogue, particularly "Would I have the temerity to do this if my bosom chum was encased therein?" It's got Freleng's excellent '40s animation team and Paul Julian's backgrounds ("Hotel Friz"). And it just strikes me as the perfect mid-point between all the different stages of Warner Brothers cartoons: it has the wackiness and pop-culture references of the early '40s, combined with the sharp gags (but not just blackout gags like in some of the '50s cartoons; they're integrated into the story) and explosions and more socially-conscious Bugs of the post-war period.
Freleng and Maltese were an exceptionally good team when it came to Bugs Bunny; they were on the same wavelength when it came to the character, both believing that Bugs shouldn't go looking for trouble but needed to be ruthless once the trouble started, and that he should go up against characters who are at least reasonably threatening. (The common thread among most of Freleng's Bugs villains, culminating in Yosemite Sam, is that while they're too stupid to be much of a threat to Bugs, other people find them dangerous and you get the feeling that they were up against anyone other than Bugs, they might not be as inept.) In this cartoon, Bugs isn't smug like he'd later become, and he's not obnoxious like he is in some of the Clampett or McKimson Bugs Bunnies. He's a likeable force for good and he's a bad-ass who enjoys inflicting violence and extremely intelligent and resourceful. And more than any other cartoon character, he takes us into our confidence: what makes the "give it to me!" joke is the way he looks at us for a split second, as if to share our enjoyment of what's about to happen. He's a fully-rounded character like few other cartoon superstars.
If the cartoon has a flaw is that there's a story gap -- we never do see how Bugs gets rid of the Peter Lorre caricature, Hugo. Knowing nothing about it, I'd guess that a scene like that was probably written but that they didn't have time to animate it. (The cartoon is almost eight minutes as it is.) Still, it actually works all right because by this point, Bugs's abilities are so well-known that we just accept that he'll get out of any situation; we don't even need to see him do it.
Personal note: back when I was in college, I organized a sort of video festival of 15 Bugs Bunny cartoons. About eight people showed up, but I think they had a good time. Of the 15 I selected, "Racketeer Rabbit" is the only one that hasn't been on a Looney Tunes Golden Collection yet, but hopefully the series will last long enough for that to change. (And it will be a bonus on a Gangster movies collection later this year.) For the less-than-eight people who care, here are the 15 Bugs Bunny cartoons I picked as a representative "best-of" chronological sampling of his career. Actually except maybe for the last two I'd probably pick the same ones today.
A Wild Hare
Bugs Bunny Gets the Boid
Little Red Riding Rabbit
The Old Grey Hare
Bugs Bunny Rides Again
High Diving Hare
Rabbit of Seville
Duck! Rabbit, Duck!
Ali Baba Bunny