Cooking chicken breasts in Buffalo sauce, cast iron
You could just add water instead of extra sauce. The water will cook off just as much as the sauce would have, leaving you with essentially just sauce by the end. If that's not spicy/saucy enough for you, then it means you did actually need some of that extra sauce you've been adding.

Alternatively, you could try lower temperature. If it's all cooking down and drying, you're presumably boiling it, but it only needs to be simmering to cook the chicken. If you're trying to get some browning, you can always increase the temperature at the end. It will probably take a bit longer to cook this way, though.

All that said, I know you said you want to use this pan, but it also kind of sounds like the pan might be too big. Extra surface area means liquid boils off faster. If it were crowded with chicken and sauce in the gaps, then it wouldn't boil off as quickly, and there wouldn't be room for sauce to dry onto the pan.