In a large skillet or saucepan, melt the butter and olive oil over medium heat. Add the onion and sauté until translucent, 3-4 minutes.
Add the garlic and cook for about one minute, stirring, until fragrant.
Combine the flour, broth, and milk in a blender (or use a handheld whisk and a bowl or liquid measuring cup) and blend until smooth.
Pour the mixture into the pan and cook over medium heat, whisking constantly, until the mixture simmers and thickens, 2-3 minutes.
Stir in the cooked chicken, salt and pepper. Bring to a gentle simmer over medium heat and cook for 3-4 minutes, stirring often.
Season to taste with additional salt and pepper, if needed.
Serve over rice with toppings such as: olives, tomatoes, shredded sharp cheddar cheese, green onions, mandarin oranges and chow mein noodles.
Notes
Chicken: the meat of a rotisserie chicken, cubed or shredded, works great here, as does canned chicken, if you prefer to use that. If you want to use uncooked chicken, use about 1 1/2 pounds boneless, skinless chicken breasts. Cut them into bite-size pieces and saute them until cooked through in step #1 with the onions. Original Recipe: over the years, I've changed how I make this recipe to make it even easier. If you are looking for the original recipe, it called for melting 3 tablespoons butter, cooking the onion and raw chicken pieces until the chicken is cooked through, adding the garlic and 1/4 cup flour and cooking for a couple minutes before whisking in the broth and milk and simmering until thickened.