Preheat the oven to 425ºF. Spray a 9×13-inch baking dish with nonstick cooking spray.
Combine 1 cup of the cheese and the chicken. Place some of the mixture down the center of each tortilla and roll up, placing in the prepared dish, seam side down.
From here you can make a sour cream and chiles sauce by proceeding to the next step, or use canned enchilada sauce and skip to step 5.
Melt the butter in a skillet over medium heat. Stir in the flour and cook for 1 minute. Slowly whisk in the chicken broth. Cook the mixture until thickened. Stir in the sour cream and the green chiles.
Pour the sauce over the top of the enchiladas. Sprinkle the remaining cheese over the top.
Bake in the preheated oven for 20 to 25 minutes, or until the cheese is melted and bubbly. Sprinkle with cilantro and cotija before serving, if desired.
