In a large, deep skillet with a tight fitting lid, heat 2 tbsp of the butter. Once it's melted, add in the peppers and onion with a pinch of salt. Cook, stirring often, until they're softened...about 5-7 minutes. Then add the garlic and the additional tbsp of butter and cook another minute or so more, until fragrant.

Turn the heat up to med-high and stir in the rice and seasonings. Let the rice toast for 3-4 minutes, stirring often. Then stir in the diced green chiles, broth, lime juice and sour cream until it's all integrated. Place the lid on and let it come to a low boil. Once at a boil, lower the heat to low and let it simmer for 15-20 minutes, or until the rice is tender and the liquid has absorbed.

Remove the pan from the heat and sprinkle the cheese across the top. Place the lid back on and let the cheese melt, then serve.