In a skillet or sauté pan; heat the oil over medium-low heat; add the onions and garlic and sauté until tender and translucent; about 3 - 5 minutes. Add ¾ of the chopped chiles and sauté for a minute longer; then add the chicken. Stir to combine. Sprinkle the flour, cumin, oregano, and a pinch of salt over and cook, stirring for a minute or two to get rid of the raw flour flavor and incorporate the flour thoroughly into the mixture (it will appear very dry, don't worry). Now slowly add the chicken stock, while stirring; it will bubble and thicken. Let it cook for just a few minutes if it gets too thick add a little additional stock. You want the meat to be moist. Taste and adjust seasonings.
While the chicken is cooking, heat the tortillas in the oven and shred and crumble the cheeses. Set-aside.
Open the sauces and pour into a bowl; stir in the remaining chiles.
Heat the oven to 375°F.
Take a baking dish large enough to hold the tortillas once they medium-low; put a thin layer of sauce on the bottom. In the center of each tortilla place about 3 tablespoons (or as desired) of chicken mixture; then top with a bit of Monterey Jack and a sprinkle of Cotia. Roll and space the enchilada in the sauce, seam side down. Continue until you have used all the tortillas and chicken. (Some of each cheese SHOULD be remaining.)
Pour the remaining sauce over the enchiladas, taking care to cover them fully (otherwise you get really crunchy ends, if you like that then no worries, leave the dry); sprinkle the remaining cheese over the top and place in the oven. Cook, uncovered for 20 minutes, or until hot and bubbly. Place briefly under the broiler to brown the cheese. Let sit for 5 minutes, then serve on warm plates.