Add 2½ tablespoons sesame oil to the pan, along with the garlic and carrots. Cook until carrots begin to soften, about 3-4 minutes. Add the thawed broccoli florets and drained water chestnuts, along with the white wine. Stir and cook until wine reduces by about half, about 3-4 minutes.
Add honey, sesame seeds, ground ginger, and soy sauce. Add the cooked chicken back in, and stir until well combined. If desired, add corn starch to thicken sauce (optional). Serve stir-fry over brown rice or quinoa. Top with fresh, chopped scallions (optional).