Preheat oven to 350°F.
Spray a 13X9 inch oven safe casserole dish with non-stick cooking spray.
In a mixing bowl combine shredded chicken with half the shredded cheese.
Divide chicken/cheese mixture evenly between the flour tortillas.
Roll tortillas snugly enclosing the the chicken/cheese filling and place them in the prepared casserole dish seam side down. Set aside.
In a large saucepan, heat butter over medium low heat until melted.
Whisk in flour and stir to combine making a roux. Cook for 1 minute.
Add chicken broth and increase heat to medium. Bring sauce to a low simmer. Continue to whisk until the sauce thickens.
Remove sauce from heat and allow it to cool for 5 minutes.
Stir in the sour cream and whisk to combine with the sauce until smooth and creamy.
Add diced green chilies to the cream sauce. Stir to combine.
Spoon the cream sauce over the enchiladas in the casserole dish. Use a spoon or a spatula to pull enchiladas apart making sure the sauce gets down in between the enchiladas.
Top enchiladas with an even layer of the remaining shredded cheese.
Bake uncovered for 20-25 minutes until heated through and cheese is melted.
Turn on broiler and cook for an addtional 2-3 minutes until cheese is golden brown and casserole is hot an bubbly.