Air carrier Captain reported a near miss with a general aviation aircraft in climb; while in contact with ATC. The Captain followed the TCAS RA command; the conflict was resolved and the flight continued.

Narrative

We were assigned a climb to FL230 by MIA Center and asked to expedite our climb. My First Officer was flying and we were expediting. Several moments later; at approximately XA:35; while climbing through approximately 12;000 feet we observed a target on TCAS flying towards us. I mentioned to my First Officer that he appeared to be coming right at us; and for him to prepare for an RA. A few seconds later we received an RA to climb. The Flight Directors and autopilot were selected off and my first officer flew the RA's guidance. The guidance commanded a climb in excess of 4000 FPM. He also had a visual on the target coming right at us and he initiated a right turn since the other pilot was also requesting a right turn. My First Officer observed some type of light twin pass underneath us by approximately 100 feet. The lateral separation was also approximately 100 feet. Automation was restored and the flight continued without further incident. I'm not sure what we could've done differently.