In a skillet or saute pan, melt the butter over medium high heat on the stovetop. When butter is melted, add diced onions and saute for 3-4 minutes until tender and slightly translucent.
Sprinkle the flour over the sauteed onions and whisk to combine. Continue to cook, while whisking for another 2 minutes.
Add in green chilies and chicken broth and whisk to combine together. Bring the mixture to a boil and continue to cook, while whisking for another 3 minutes until the sauce starts to thicken.
Remove the sauce from the heat and whisk in sour cream, salt, and pepper.
Ladle approximately ½ cup of the sauce into a 9x13 baking dish, just to cover the bottom.
In a medium bowl, mix together shredded chicken and 1 ¼ cups of the shredded cheese. Place about ⅓ cup of the chicken and cheese mixture into the center of each flour tortilla. Roll up each tortilla tightly and place seam side down into the baking dish.
Pour the remaining sauce over the top of the tortillas and top with the remaining shredded cheese.
Cover the pan with foil and bake the enchiladas in the preheated oven for 20 minutes or until cheese is melted and enchiladas are heated through.