I second ultrasonics. Basically own one for suppressor cleaning, so same thing: carbon, carbon, carbon. I don't do most of my brakes because they are on rifles and my ultrasonic is too small, but for a pistol, totally would immerse this and let it ride.
So many notices about Simple Green attacking aluminum and some plastics I don't even have it in the shop. There are ultrasonic cleaners for guns that are safe and super effective, so get a bit of that along with it. Hot water. If the cleaner doesn't heat, or heat much, get the hottest thing that comes from your tap to start with for the water part.
It may take a couple rounds. Pull and scrub with a toothbrush, wipe away residue (much will now wipe, but not all falls off) then re-immerse until clean.