Preheat the oven to 350°F (175°C) and grease a 9x13-inch baking dish.
For the filling, melt the butter in a large skillet over medium heat.
Add the minced onion, minced garlic, and chopped mushrooms and sauté until the mushrooms are soft and liquid has mostly evaporated, about 5–7 minutes.
Remove about half of the mushroom mixture from the skillet and set it aside to use in the sauce.
To the remaining mushroom mixture in the skillet, add 1/4 cup chicken broth and cook for 2 minutes.
Add the chopped spinach and cook until wilted and reduced, then stir in the softened cream cheese until melted and combined.
Stir in the cooked chicken and season with salt to taste; remove from heat.
For the sauce, combine the reserved mushroom mixture, lime juice, 3/4 cup chicken broth, garlic powder, sour cream, and chopped cilantro in a small saucepan.
Simmer the sauce over low heat, stirring, until it just begins to thicken. Taste and add salt as needed.
Divide the filling evenly among the 8 flour tortillas, roll them up, and place seam-side down in the prepared baking dish.
Pour the sauce evenly over the rolled enchiladas, sprinkle the shredded Monterey Jack cheese on top, and bake for 10–15 minutes until bubbly and the cheese is melted.