Preheat oven to 350 degrees F (175 degrees C). Lightly grease a large baking dish.
Melt the butter in a skillet over medium heat. Cook and stir green onion and garlic until fragrant, about 2 minutes. Add mushrooms and cook until lightly browned and tender, about 6 minutes. Stir in green chilies, cream of mushroom soup, and sour cream. Mix well. Reserve 3/4 this sauce in a bowl and set aside. Mix in chicken, pinto beans, and 1/2 cup of shredded Cheddar cheese to the remaining 1/4 of sauce in saucepan; stir together.
Fill each tortilla with chicken mixture and roll up. Place seam side down in the prepared baking dish.
Combine reserved 3/4 sauce with milk in a small bowl. Spoon this mixture over the rolled tortillas and top with remaining 1/2 cup shredded Cheddar cheese. Bake in the preheated oven until cheese is bubbly and enchiladas are heated through, 30 to 35 minutes.