Melt butter in a Dutch oven over medium heat; add onion, poblano chile, and garlic, and saute until tender. Add rice, and cook, stirring constantly, 5 to 7 minutes. Add 1 cup chicken broth; cook, stirring constantly, until liquid is absorbed.
Repeat procedure with remaining broth, 1/2 cup at a time. (Cooking time is about 1 hour.) Stir in crawfish; cook, stirring constantly, 4 minutes. Add Monterey Jack cheese and Creole seasoning, stirring until cheese melts. Garnish, if desired.
Prep: 20 min.; Cook: 1 hr., 15 min.
*2 pounds cooked, peeled, medium-size shrimp may be substituted for crawfish.