Preheat the oven. Set to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Sauté the onion. Warm olive oil in a skillet over medium heat.
Cook the diced onion for 3–4 minutes until softened and translucent. Remove from heat.
Mix the chicken filling. In a large bowl, combine shredded chicken, sautéed onion, diced green chiles, cumin, chili powder, garlic powder, salt, and pepper. Stir in 1/2 cup of enchilada sauce to moisten.
Blend or stir the cottage cheese. For a smoother texture, pulse the cottage cheese in a blender or food processor for 15–20 seconds.
If you like curds, skip blending.
Prepare the cheeses. In a separate bowl, mix 1 cup Monterey Jack with the cheddar cheese. Reserve the remaining 1/2 cup Monterey Jack for the top.
Soften the tortillas (optional but helpful). If using corn tortillas, wrap them in a damp paper towel and microwave for 30–45 seconds to make them more pliable.
Start layering. Spread 1/2 cup enchilada sauce over the bottom of the baking dish. Add a single layer of tortillas, slightly overlapping.
Add the fillings. Spread half the chicken mixture over the tortillas.
Spoon half the cottage cheese over the chicken and gently spread. Sprinkle with half of the cheese blend. Drizzle with 1/2 cup enchilada sauce.
Repeat the layers. Add another layer of tortillas, the remaining chicken, the remaining cottage cheese, most of the remaining cheese blend, and another 1/2 cup sauce.
Top it off. Finish with a final layer of tortillas, the rest of the enchilada sauce, and the reserved Monterey Jack.
Bake. Cover loosely with foil and bake for 20 minutes.
Remove foil and bake another 10–12 minutes, until the cheese is bubbly and lightly golden.
Rest and garnish. Let the casserole rest for 10 minutes before slicing. Top with cilantro, green onions, and any extra toppings you like.