Sauté onion in hot oil over medium-high heat 5 minutes or until tender.
Stir together chicken, 1/4 cup sour cream, and 1 1/2 cups shredded Monterey Jack cheese. Stir in onion. Spoon chicken mixture evenly on one end of each tortilla, and roll up. Arrange, seam side down, in a lightly greased 13- x 9-inch baking dish.
Whisk together 2 cans green chiles, milk, and flour in a saucepan over medium heat; gradually add remaining 2 1/2 cups cheese, stirring constantly, until cheese is melted and mixture is smooth; pour sauce evenly over top of enchiladas. Spread remaining sour cream over sauce.
Bake enchiladas at 400° for 20 to 25 minutes or until bubbly. Drain remaining can of green chiles, and spread evenly on top of enchiladas.