Preheat oven to 375° Fahrenheit. Spray (1) 9x13 and (1) 8x8 pan with non stick cooking spray.
In a skillet over medium heat, melt 2 Tbsp butter. Add onion and garlic. Saute until translucent. Mix in green chiles. Remove from heat. Add in shredded chicken and cheese. Mix well.
Spoon about 1/4 C of chicken onto each tortilla. Tightly roll tortilla and place seam side down in baking dish. (Tortillas roll best when chicken is placed more to the side than down the center of the tortilla.) Set enchiladas aside. If there is any leftover filling, save it to add to the sauce.
Sour Cream Sauce
In a skillet over low heat, melt 3 Tbsp butter. Add flour and use a whisk to blend all of the flour and butter together until thick and smooth. Pour in broth and whisk again until smooth. Whisk in sour cream until smooth and creamy. Add any leftover filling to sauce. Remove from heat and pour over enchiladas.
Top enchiladas with cheddar cheese. Bake for 20-25 minutes until cheese is melted and sauce bubbling around the edges.