Preheat the oven to 375°F with a rack in the center position.
In a large bowl combine the shredded chicken, 1½ cups of the mozzarella, 1½ cups of the Monterey jack, sour cream and taco seasoning.
Wrap the tortillas in a damp paper towel or tea towel and microwave for 1 minute.
Warm the enchilada sauce in a small skillet over medium-low heat. Working with one tortilla at a time, dip it in the enchilada sauce until lightly coated on both sides. Scoop ⅓ cup of the enchilada filling into the tortilla and roll tightly, arrange (seam side down) in a 9x13 baking dish. Repeat with remaining tortillas and filling.
Pour the remaining sauce over the prepared enchiladas. Sprinkle all over with the remaining mozzarella and the remaining Monterey Jack.
Bake for 25 minutes, or until the sauce is bubbling and the cheese is melted. Let rest for 5 minutes. Sprinkle with cilantro and red pepper flakes, if using, before serving.
Notes
To make-ahead, assemble the enchiladas, cover, and refrigerate up to 24 hours before baking.
To freeze, allow the baked enchiladas to cool to room temperature. Cover tightly with a lid or plastic wrap, followed by foil and place in the freezer for up to 3 months.