I don't even know if such a place exists, but there used to be a little Yakatori bar in the Mid Wilshire area. They had these wonderful little skewers with all different stuff and they had what they called a "stop-course". They would just keep them coming until you told them to stop and you were charged per skewers. They closed years ago.
Anyone know of any place like this? Westside or Valley?
It's been years since I've been, but you might try Sakura House. It was quite good then, but I can't speak to its current state.
13362 Washington Blvd
Sakura House falls into the broader category of kushiyaki that Jerome describes. I hate to correct my most esteemed chow colleague, but kushikatsu is a specific dish: a fried, breaded pork cutlet on a skewer. I think he meant to say the style of grill cookery called robatayaki.
The place you described sounds like Yoriki which was on 3rd St.
The best place i know of these days is Shinsengumi, on Western in Gardena
Had a good meal recently at an old standby - Nanbankan in West LA. You sit at the bar which goes all around the grill and watch the chefs seasoning and flipping the skewers. Fun and tasty.
My favorite is in Little Tokyo:
Weller Court, 2nd Floor
I've been going for years and they've always been great. Mostly Japanese crowd. They start you off with cabbage & carrots and a tasty miso dipping sauce, then you start ordering skewers. Highly recommend the tsukune (loosely packed chicken meatballs) and the kaori (pork belly and shiso leaf). It's also nice to get a grilled rice ball as a closer. Worth the trip downtown.