In a large saucepan, melt butter over medium-high heat. Add in mushrooms and cook until mushrooms are soft, about 5 minutes.
Add in flour and stir until there are no more specks of flour left. Pour in chicken broth and milk and bring to a boil. Reduce heat to low and simmer until sauce is thickened, about 3 minutes.
For a richer sauce: in a small mixing bowl, whisk together egg yolks with heavy cream. Working quickly, slowly pour in 1/2 cup of the hot mixture into the egg mixture while whisking vigorously. Immediately pour this mixture back into the saucepan, whisking the entire time. Cook 2 minutes more.
Stir in frozen peas, drained pimientos, and cooked chicken and let heat through, 2 to 3 minutes.
Serve hot over cooked rice, pasta, toast, or biscuits.
Chicken a la King is served over rice, pasta, toast, or biscuits.