In small bowl, combine broccoli and cheese; place mixture at one end of each chicken breast; roll up and secure with toothpicks.
Place flour in shallow dish; salt and pepper to taste. Place egg in second shallow dish and bread crumbs in third shallow dish. Dust each chicken breast with flour, dip in egg and roll in bread crumbs.
In large skillet, heat oil and butter over medium heat. Add chicken and sauté, turning occasionally until all sides are browned (about 5 minutes).
Transfer to a greased baking dish; bake in a preheated 375°F oven 30 minutes until internal temperature of chicken reaches 170°F and juices run clear.