Butcher in LA??
There really is no L.A. equivalent of Florence Meat Market. European Sausage Kitchen in Beverly Hills has the small, old world thing down, but obviously sells sausage instead of steak; the disarmingly Teutonic Schreiner's in the Montrose/Glendale area has great pork things, but no steak. (There is, I think, a place in San Marino that may or may not be called San Marino Meats that is pretty much what you seem to be looking for.)
Bristol Farms, though, does have a very nice meat department, and swell aged steaks. Superior Meat, on Pico near Beverly is also good, and also ages its steaks.