Preheat oven to 400°. Spray 13x9-inch baking dish with cooking spray; set aside.
Heat a cast-iron or non-coated skillet over medium-high heat until hot. Toast dried peppers in skillet, turning occasionally, until they puff up, about 3 minutes. Place peppers in a small bowl and cover tightly with plastic wrap. Place garlic cloves and onion quarters, cut side down, onto skillet and char all sides. Place charred onion and garlic in a medium sauce pot. Cut toasted peppers in half; remove stems and seeds. Roughly tear or slice peppers and add to pot; add water and simmer over medium heat for 15 minutes.
Reserve 1/2 cup of the simmering liquid, then drain pepper-onion mixture. Place pepper-onion mixture in a blender. Add tomato sauce, cumin, salt, oregano, paprika and sugar; blend until smooth. Using a fine-mesh sieve or several layers of cheesecloth, strain pepper mixture well (note: if using a fine-mesh sieve, a ladle works well to press the sauce through the sieve). Return strained sauce to sauce pot; stir in reserved simmering liquid and chicken stock. Place chicken in sauce pot, cover and cook over medium heat 10 to 15 minutes until chicken breasts are cooked through (165°F). Remove chicken and cool slightly. Shred chicken, combine with 1/4 cup cotija, 1/2 cup mozzarella, and 1/4 cup of the sauce.
Wrap tortillas in damp paper towels and place on microwave-safe plate. Microwave on HIGH 1 minute or until tortillas are warmed and softened. Top each tortilla with about 1/4 cup of the chicken-cheese mixture; roll up. Place tortillas seam-side down in baking dish. Pour sauce over top, then sprinkle with remaining cotija and mozzarella cheese. Cover with aluminum foil and bake for 15 minutes or until cheese is melted and sauce is bubbling. Remove from oven and allow to cool slightly. Garnish with fresh cilantro; serve.
Make sure to wear latex or plastic gloves when removing seeds from dried peppers. If desired, serve enchiladas with a few drizzles of crema or a dollop of sour cream.