Whisk the chicken broth, soy sauce, cornstarch, and ginger together in a small bowl; reserve.
Heat oil in a large skillet or wok. Cook and stir chicken and garlic in the oil until chicken is cooked through, 8 to 10 minutes. Stir in mushrooms, water chestnuts, and reserved chicken broth mixture. Cook until sauce begins to thicken, 3 to 5 minutes.
Stir snow peas into the pan and cook until tender, 3 to 5 minutes. Transfer to a platter and sprinkle with sesame seeds before serving.