In a frying pan, cook chicken and onion together in oil over medium-high heat until chicken is just done.
Divide cooked chicken evenly between tortillas; add cheese to each tortilla.
Roll enchiladas and place seam-side down in a square baking dish.
Melt butter in a medium saucepan; stir in flour to make a roux, stir and cook until bubbly, and gradually whisk in chicken broth then bring to boiling, stirring frequently.
Remove from heat; stir in sour cream and green chilies.
Pour sauce evenly over enchiladas.
Top with remaining 3/4 cup cheese and bake at 400° F for 20 minutes until cheese is melted and sauce near edges of baking dish is bubbly.