In a skillet or pan, add 2 tbsp oil and bring to heat over a medium-high burner.
Add mushrooms and cook until quite darkened (all of the moisture should be driven off).
Add onion / pepper and a generous amount of salt, and cook until softened, around 5 minutes.
Add the garlic, cooking for 30s.
Add the chicken and 1/2 of the cheese, cooking until cheese is melted.
Meanwhile, in a small saucepan, make a roux by melting butter and adding flour. Cook flour for ~30s to take off the raw edge.
Add cold milk slowly, stirring to prevent lumps from forming to make a bechamel.
Add the salsa verde and bring to simmer, then remove from heat.
Lightly oil a baking dish, then fill tortillas with chicken mixture and place in rows in the dish.
Cover the enchiladas with the milk/salsa mixture, then top with cheese.
Bake for 30-40 minutes in a 400° oven, taking care not to burn the cheese (may have to add foil at first). Once cooked through and bubbling, broil the top to caramelize the cheese topping.
Serve with refried black beans and cilantro lime rice.