Make the enchilada sauce up to three days ahead; the flavors will meld and only get better.
Prepare Sauce: Preheat broiler with oven rack 5 inches from heat. Broil first 5 ingredients on a foil-lined baking sheet, 6 minutes or until peppers blister and char. Remove from oven; let stand 10 minutes or until cool enough to handle. Peel and seed peppers; peel garlic. Reduce oven temperature to 375°.
Bring tomato sauce, next 5 ingredients, broiled vegetables, and 1 tsp. salt to a boil in a large saucepan over high heat, stirring occasionally. Reduce heat to medium-low; cook, stirring occasionally, 35 minutes. Remove from heat; cool 10 minutes. Process with a handheld blender or in batches in a food processor or blender until smooth.
Prepare Enchiladas: Shred 2 oz. Monterey Jack cheese to yield 1/2 cup. Cut remaining Monterey Jack cheese into 12 (4- x 1/2-inch) sticks.
Spread 1 cup enchilada sauce in a 13- x 9-inch baking dish coated lightly with cooking spray. Spread a thin layer of sauce on 1 tortilla. Place 1/3 cup chicken and 1 Monterey Jack cheese stick on edge of tortilla, and roll tortilla. Place in baking dish, seam side down. Repeat with remaining tortillas, chicken, and cheese sticks. Pour 2 1/2 cups sauce over tortillas. Chill remaining sauce for another use. Bake enchiladas at 375° for 30 minutes.
Top enchiladas with shredded Cheddar and shredded Monterey Jack cheese; bake 5 more minutes or until cheese is melted. Remove from oven; let stand 10 minutes.
Meanwhile, combine avocado, next 2 ingredients, and remaining 1/2 tsp. salt; serve with enchiladas.