Why did my peanut butter sauce split?
You need to 1) increase the emulsification, and 2) reduce the amount of time the sauce is very hot.

You can try adding honey or mustard to the sauce, that will improve emulsification. You can also hit it with a stick blender which will do a much better job of breaking it up than can be done by hand.

Also it may look ok after a few hours but who knows how much separation is starting to happen at a very small scale and will accelerate when it hits heat.

Add the sauce at the last possible moment before plating. It isn't going to benefit from cooking, and it isn't going to marinade into the cooked food, so don't have it in any longer than it takes for it to warm up. 20 seconds is probably all it will take.