Melt butter in 12-inch skillet until sizzling; add chicken pieces. Cook over medium-high heat until chicken is lightly browned (4 to 6 minutes). Add mushrooms; continue cooking until chicken is no longer pink and mushrooms are tender (4 to 6 minutes). Do not drain.
Spread 1/2 cup sauce into greased 13x9-inch baking dish. Arrange single layer of frozen ravioli over sauce; drizzle 3/4 cup sauce evenly over ravioli. Spread with 1 1/2 cups chicken and mushroom mixture, 1/2 cup red bell pepper and 1 cup mozzarella cheese. Repeat with remaining ingredients, except remaining 1 cup mozzarella cheese and Parmesan cheese. Cover tightly with aluminum foil.
Bake for 45 minutes.
Remove foil; sprinkle with remaining mozzarella and Parmesan cheese. Bake, uncovered, for 15 to 20 minutes or until cheeses are melted. Let stand 15 minutes before serving.
TIP: To make ahead, prepare as directed above except do not bake. Cover; refrigerate up to 24 hours. When ready to bake, continue as directed above. Increase first baking time to 1 hour.