In a large saucepan, melt butter. Add mushrooms, onion, celery and red bell pepper. Cook for about 3 minutes, or until vegetables are tender.
Stir in flour, poultry seasoning, 1/2 teaspoon salt (less if using salty broth) and 1/4 teaspoon pepper. Add the broth and milk; cook and stir until thickened and bubbling.
Stir in chicken and peas; pour into a round casserole dish. Brush the edges of the casserole with beaten egg to help the pie crust adhere.
Place pie crust over the mixture; cut to fit and crimp edges all around. Cut steam vents in top.
If desired, re-roll and cut excess pastry into strips or decorative shapes and lay over the crust. Brush chicken pot pie with remaining beaten egg.
Bake at 400° for 30 to 40 minutes, or until crust is browned.