Melt the butter in a large pan. Saute the onions till tender, then add the garlic for the last minute or two.
Whisk in the salt, pepper, and flour. Slowly whisk in the milk and chicken broth. Cook and stir over medium heat till bubbly. Boil for at least one minute, stirring constantly.
Add the shredded chicken and heat through.
Serve over hot white or brown rice with your choice of toppings.
Notes
If you don't have cooked chicken on hand, you can cook 1 1/2 pounds of boneless, skinless chicken breast.
The sauce is also tasty with 1/2-1 cup of sour cream stirred in with the chicken.