Preheat the oven to 400°F Grease a 10" round glass casserole dish.
Rinse broccoli in cold water to thaw; drain thoroughly.
Mix together the broccoli, carrots, 1 1/3 cups cheese, chicken, and onion right in the casserole dish.
Beat milk, eggs, Bisquick, salt and pepper until smooth. Pour on top of the chicken mixture that is in the casserole dish.
Bake 30 to 35 minutes or until a knife inserted into the center comes out clean. Top with remaining cheese. Bake another couple of minutes until the cheese melts.