To prepare filling, place first 6 ingredients in a Dutch oven; bring to a boil. Cover, reduce heat, and simmer 45 minutes. Remove chicken from cooking liquid; cool. Remove chicken from bones; shred with 2 forks. Discard bones. Reserve broth for another use.
Preheat oven to 375°.
Combine chicken, 1/2 cup each Monterey Jack and cheddar cheeses, and 1/2 cup onion in a bowl; set aside.
To prepare sauce, place milk and next 5 ingredients (milk through chiles) in a food processor; process until smooth.
To prepare tortillas, fill a medium skillet with 1 inch of water; bring to a simmer. Dip 1 tortilla in water using tongs. Spoon 1/2 cup filling in center of tortilla; roll tightly, and place in an 11 x 7-inch baking dish. Repeat procedure with remaining tortillas and filling.
Pour sauce over enchiladas. Cover and bake at 375° for 20 minutes. Uncover; sprinkle with 1/4 cup each Monterey Jack and cheddar. Bake an additional 5 minutes or until cheeses melt. Top with sour cream.
According to one online reviewer, "If you're thinking enchiladas, try this one!" The recipe is extra cheesy with both Monterey Jack and Cheddar, but it will also be fine if you choose to use only one type of cheese.