Cook the chicken using your preferred method—boil, bake, or use a rotisserie chicken.
Once the chicken is cooked, preheat the oven to 350°F and grease a 9x13-inch baking dish.
Shred the cooked chicken with a fork if needed, and add it to a bowl. Mix in 3/4 cup of shredded Monterey Jack cheese or Pepper Jack cheese, garlic powder, and cream cheese. Stir well to combine.
In a saucepan, melt the butter, then stir in the flour and taco seasoning. Cook for about 1 minute. Add 2 cups of chicken broth, whisking until smooth. Stir in 1/2 cup of shredded cheese and cook over medium heat until the sauce thickens and bubbles.
Stir in the sour cream and green chilies, making sure not to bring it to a boil. Remove the saucepan from the heat.
Spoon some of the chicken and cheese mixture into each tortilla, roll it up, and place the rolled tortillas in the baking dish.
Pour the sauce over the enchiladas and top with the remaining shredded cheese.
Bake for 22 minutes, then broil on high for an additional 3 minutes to brown the cheese.