Preheat your oven to 350ºF. Grease a 9 x 13 pan to prevent sticking.
In a skillet, melt 2 tablespoons of butter over medium heat. Add chopped onions and bell peppers. Sauté until tender.
Combine sautéed vegetables with shredded chicken and green chilies. Stir and set aside.
Melt 3 tablespoons of butter in the same skillet. Add flour and stir to form a roux. Cook until lightly golden.
Gradually whisk in chicken broth, stirring until sauce thickens and bubbles. Remove from heat.
Stir in sour cream, ½ cup Monterey Jack cheese, and ¼ cup sauce into the chicken mixture.
Pour some of the sauce into the bottom of the greased pan.
Fill each tortilla with chicken mixture, roll up, and place seam-side down in the pan.
Pour remaining sauce over the filled enchiladas and sprinkle with remaining cheese.
Bake for 25 minutes until cheese is bubbly and lightly golden. Enjoy!