Preheat oven to 350F degrees. Spray 9x13-inch baking dish with nonstick cooking spray.
In a medium-sized bowl, mix together 2 cups shredded, cooked chicken with 1 cup shredded Monterey Jack cheese along with salt, pepper, Adobo seasoning, to taste.
Place chicken mixture into each of the 10 flour tortillas.
Roll them all up and place them in the prepared baking dish.
In a medium sauce pan, melt 3 Tablespoons butter. Whisk in 3 Tablespoons all-purpose flour and let cook and thicken for 1 minute (don't let it burn).
Add 2 cups chicken broth and whisk until smooth.
Stir in 1 cup sour cream and 4 ounce can diced green chiles. Be sure not to let the mixture boil.
Take sauce off the heat and pour it over the enchiladas.
Top with 1 cup shredded Monterey Jack Cheese.
Bake for about 20-25 minutes. If you'd like, you can turn the broiler on high and broil these for a minute or two until cheese is slighly browned and bubbly.