A combustor assembly for a gas turbine engine includes a combustor dome defining an opening, a cooling hole, and at least in part defining a combustion chamber. The combustor dome includes a first side and a second side, the cooling hole extending from the first side to the second side. The combustor assembly additionally includes a fuel-air injector hardware assembly positioned at least partially within the opening of the combustor dome and including a heat shield. The heat shield includes a heat deflector lip. The cooling hole in the combustor dome is oriented to direct a cooling airflow onto the heat deflector lip to maintain at least a portion of the heat shield within a desired operating temperature range.