Melt butter in a saucepan over medium heat; cook and stir green onions and garlic in butter until fragrant and softened, about 2 minutes. Stir chicken, spinach, chili powder, black pepper, and salt into the green onion mixture and cook until heated through, about 5 minutes. Remove from heat. Mix ricotta cheese, sour cream, 1 cup Monterey Jack cheese, and 1 cup of mozzarella cheese into chicken mixture until thoroughly combined. Reserve remaining Monterey Jack and mozzarella cheeses for topping.
Place tortillas into a plastic bag, seal bag, and microwave on high until warm and soft, 30 to 45 seconds. Spoon about 3/4 cup of chicken filling down the center of each tortilla in a line. Roll tortillas around filling and place with seam sides down into a 9×13-inch baking dish.
Pour enchilada sauce over filled enchiladas and sprinkle the remaining 1/2 cup of Monterey Jack and mozzarella cheeses over the top.
Bake in the preheated oven until filling is bubbling and cheese topping is browned at the edges, about 30 minutes.