I don't know St.Louis well, but I have to stay over there fairly regularly and I usually find somewhere to eat in the area known as "The Hill". It is an Italian/Sicilian part of the city with a lot of restaurants, most you can just walk in to. But the best are probably
Giovanni's and
Tony's (one of the best restaurants in the country). They are both quite expensive and if you did want to try either of them, you would need to reserve, today or tomorrow at the latest.