Peel, stem and seed chiles, then finely chop (mince Anaheims and poblanos, since they're sturdier). Preheat oven to 400°.
Heat oil and butter in a large frying pan over medium heat. Add garlic and cook until fragrant, about 30 seconds. Stir in chiles, salt, and pepper. Cook, stirring occasionally, 3 minutes to blend flavors. Add 1 1/4 cups broth and simmer until reduced by one-third, about 10 minutes.
Meanwhile, prepare tortillas: In a small frying pan, bring remaining 2 1/2 cups broth to a gentle simmer. Working with one at a time, very briefly dip tortillas into broth to barely soften. Transfer each tortilla to a large baking sheet (you may need 2 or 3 sheets). Do not overlap or tortillas will stick.
Divide 1 1/4 cups cheese equally among tortillas and top each with shredded chicken, dividing evenly. Wrap tortilla around filling and transfer, seam-side down, to a 9- by 13-in. baking dish.
Pour chile sauce over enchiladas, leaving an inch or so bare at either end of the enchiladas if you like a bit of crunch, and top with remaining 3/4 cup cheese. Bake until cheese is bubbling and browned, 15 to 20 minutes. Serve with sour cream.
*Find frozen flame-roasted New Mexico green chiles, mild to spicy, for $6 for 5 lbs. at newmexicanconnection.com. If using Anaheims and poblanos, broil until they blacken all over, 10 to 15 minutes, turning once, and let them cool before peeling.
Cooking with chiles
This recipe is best when made with New Mexico chiles, preferably northern varieties such as Chimayo. Anaheim chiles, which are a New Mexico variety, are widely available throughout the West and make a fine substitute for northern green chiles—roast them over a stovetop burner or under a broiler to blacken the skins. (And if you're sensitive to chiles, wear gloves when handling.) Canned green chiles just don't cut it here.
See "Finding New Mexico Chiles," (below) for mail-order sources.
Finding New Mexico chiles
Native Seeds/SEARCH. Grow northern New Mexico chiles from heirloom seeds. www.nativeseeds.org or 866/622-
New Mexican Connection. We couldn't find a reliable mail-order source for roasted northern green chiles, but we did find good roasted Sandia chiles here. $56 for 5 lbs., including shipping; www.newmexicanconnection.com or 800/933-
Santa Fe Farmers Market. The best place to find northern New Mexico chiles, both fresh and dried. Various locations and hours; contact www.santafefarmersmarket.com or 505/983-