In a large skillet melt 1 tbsp of the butter over medium heat. Add onions and peppers and cook until tender. Add garlic and saute another minute.
Melt 2 tbsp butter in the onion and pepper mixture then add flour and cook and stir to make a roux.
Whisk in the chicken broth and stir until smooth, then bring to a boil.
Whisk in the milk and bring to a low simmer. Stir until thickened.
Add chopped cilantro, salt and pepper to the sauce, stir and turn off the heat.
Preheat oven to 375 degrees. Assemble the enchiladas. Pour about 1/2 cup of sauce or enough to coat the bottom of a 9 x 13 glass baking dish.
Place 1/4 cup chicken and desired amount of grated cheese on one side of the tortilla and roll up tightly.
Place the enchiladas seam side down on top of the sauce in the baking dish. Continue with the rest of the enchiladas.
Pour the remaining sauce over the top of the enchiladas and spread over them evenly.
Sprinkle shredded cheese generously on top of the sauce.
Bake, uncovered, for 20-30 minutes or until enchiladas are hot and bubbly and cheese is melted. Remove from the oven and squeeze lime juice over the top. Garnish with more chopped cilantro and serve!