Heat oven to 350°F and grease a 9x13-inch baking dish.
Melt butter in a skillet over medium heat. Soften onion until translucent, about 5 minutes.
Stir in garlic and cumin. Cook until fragrant, 30 seconds. Remove from heat.
Whisk sour cream, cream cheese, broth, and lime juice in a large bowl until smooth.
Fold in cooked rice, green chiles, sautéed onions, salt, pepper, and half of each cheese.
Spread mixture in the dish. Top with remaining cheeses.
Bake until hot and bubbling with lightly browned spots, 30–35 minutes.
Rest 10 minutes. Sprinkle with cilantro and serve warm.