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.