Start by rinsing the uncooked white rice under cool water. This helps to remove excess starch and prevents the rice from becoming too sticky. Drain well.
In a large pot or Dutch oven, combine the rinsed rice and chicken broth. Bring the mixture to a boil over medium-high heat. Once boiling, reduce the heat to low, cover, and let it simmer for about 15-20 minutes, or until the rice is tender and the liquid is absorbed.
While the rice is cooking, in a separate bowl, mix together the sour cream, shredded Monterey Jack cheese, shredded sharp cheddar cheese, and the squeezed dry chopped mild chiles. Add a pinch of onion salt to enhance the flavor.
Once the rice is cooked, remove it from the heat and let it sit for a few minutes. Then, gently fluff the rice with a fork. Add the cheese mixture to the pot and stir until everything is well combined and the cheese is melted.
You can serve the Chile Cheese Rice immediately for a creamy dish, or transfer it to a baking dish and bake at 350°F for about 15-20 minutes. Baking will give it a nice golden crust on top.