Heat the oven to 375°F (190°C). Lightly grease a 9-by-13-inch baking dish.
Heat the olive oil in a large skillet over medium heat. Add the onion and cook for 4 to 5 minutes, until softened. Add the garlic and cook for 30 seconds, just until fragrant.
In a large bowl, combine the chicken, rice, green chiles, cooked onion and garlic, cumin, smoked paprika, salt, pepper, broth, Greek yogurt, and 1 cup of Monterey Jack. Stir gently until evenly coated.
Spread the mixture evenly in the prepared baking dish. Sprinkle the remaining 1 cup Monterey Jack over the top.
Bake uncovered for 25 to 30 minutes, until the edges bubble, the cheese is lightly golden, and the center reaches 165°F (74°C) on an instant-read thermometer.
Let the casserole rest for 5 to 10 minutes. Garnish with cilantro and serve with lime wedges.