In a large skillet, heat the butter over medium heat. Add the onion, carrots and mushrooms. Cook until the vegetables are tender, stirring occasionally, about 3 to 5 minutes. Whisk in the flour and cook for 2 minutes. Reduce the heat to low. Whisk in the chicken broth and milk, and cook until the mixture boils and thickens. Stir in the finely chopped chicken, peas and sour cream. Season with the salt and black pepper. Remove the skillet from the heat.