Over medium-high heat, melt butter in an oven-safe dutch oven or very large skillet with high sides. Sauté onions, celery, and bell peppers for about 7 minutes.
Add spices and garlic, and cook for about 1 minute.
Stir in diced tomatoes and green chilies, broth, milk, rice, bay leaf, crawfish, and ½ cup cheese.
Press into an even layer and transfer to preheated oven. Cook for 30 minutes.
Remove from the oven, stir, and top with cheese. Bake for another 15 minutes until the liquid has completely absorbed and the cheese is lightly brown.