Whisk soy sauce, sugar, and cornstarch together in a bowl until smooth; stir turkey into cornstarch mixture and set aside for 15 minutes to marinate.
Heat a large skillet over high heat and pour vegetable oil into the hot skillet, coating the bottom. Stir broccoli and carrots into the hot oil to combine; pour chicken broth into the vegetables and cover skillet. Cook until carrots start to become tender, about 2 minutes.
Reduce heat to medium. Stir turkey and sauce into vegetables; cook and stir until heated through and the sauce has thickened, 1 minute. Stir mushrooms into turkey and vegetables until combined.
