Preheat your oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Warm a large skillet over medium heat.
Add olive oil, then cook the onion for 3–4 minutes until softened. Stir in the garlic and cook 30 seconds, just until fragrant.
Add the green chiles, enchilada sauce, and chicken broth. Bring to a gentle simmer.
Reduce heat to low.
Stir in the cream cheese cubes until melted and smooth, about 2–3 minutes. Whisk in the sour cream, cumin, chili powder, salt, and pepper. Taste and adjust seasoning.
Stir the shredded chicken into the sauce to coat evenly.
Turn off the heat.
Assemble the casserole: Spread a thin layer of the chicken mixture across the bottom of the baking dish. Add a layer of tortillas (either quarters or whole, depending on your style). Top with more chicken mixture and a handful of both cheeses.
Repeat the layers (tortillas, chicken, cheese) 2–3 times, finishing with a generous cheese layer on top.
Cover the dish loosely with foil and bake for 15 minutes.
Remove the foil and bake another 10–12 minutes, until the cheese is melted and bubbly with golden spots.
Let the casserole rest for 10 minutes before slicing. This helps it set and makes serving easier.
Garnish with chopped cilantro and a squeeze of lime if you like. Serve warm.