Melt the butter. Add the onion and cook until softened, stirring. Stir in the flour, stirring constantly and let cook a few minutes over medium heat. Stir in the chicken broth and milk, stirring constantly. A whisk works well for this. Heating the chicken broth and milk helps make things go smoother here.

Let this cook until simmering, stirring frequently. Simmer at the lowest possible bubbling point until thickened a bit. Stir in the chicken. Add the cheese if you want. I've made it often without any cheese but it's a nice touch if you have it. Taste and salt and pepper as needed.