In a medium-size saucepan, heat the butter over medium heat. Add the onion, and cook until soft.
Add the flour, salt, pepper, and seasonings, and stir to combine well. Cook, stirring constantly, until the mixture has just begun to turn a very light brown color.
Add the bone broth slowly, stirring constantly. Add the milk, and continue to stir until the mixture is smooth. Bring the mixture to a simmer, and continue to cook, stirring occasionally, for about 10 minutes until the mixture thickens.
Remove the saucepan from the heat, and add the cooked chicken. Stir to combine.