Preheat oven to 350°. Melt 2 Tbsp. butter in a large skillet over medium heat; add onion and bell pepper, and sauté 5 minutes or until tender.
Stir together onion mixture, chicken, 1 can diced green chiles, and 2 cups shredded cheese. Spoon a heaping 1/3 cupful chicken mixture on 1 end of each tortilla, and roll up. Arrange enchiladas, seam sides down, in a lightly greased 13- x 9-inch baking dish.
Bake at 350° for 10 minutes.
Meanwhile, melt remaining 3 Tbsp. butter in a heavy saucepan over low heat; whisk in flour until smooth. Cook, whisking constantly, 1 minute. Gradually whisk in chicken broth and milk; cook over medium heat, whisking constantly, 3 to 4 minutes or until thickened and bubbly. Remove from heat, and stir in remaining 2 cans green chiles.
Remove enchiladas from oven. Pour green chile mixture over enchiladas. Sprinkle with remaining 1 cup cheese.
Bake at 350° for 20 to 25 minutes or until bubbly. Serve with desired toppings.