A combustor includes a combustor head with an integrated counter-flow heat exchanger. The combustor head defines a combustion chamber and has a surface accessible for external delivery of thermal energy. The counter-flow heat exchanger has a fuel channel extending between a fuel inlet and a fuel outlet coupled to the combustion chamber, an exhaust channel extending between an exhaust inlet coupled to the combustion chamber and an exhaust outlet, and a shared wall between the fuel channel and the exhaust channel for transfer of thermal energy therebetween.