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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 8 tortillas; add 1 1/2 tablespoons cheese to each tortilla.
Roll enchiladas and place seam-side down in a 9x13" baking dish that has been lightly sprayed with no-stick cooking spray.
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 (a baking dish may be double-wrapped and frozen at this point) and bake at 400° F for 20 minutes until cheese is melted and sauce near edges of baking dish is bubbly.