Marinate beef in cola for at least 30 minutes and up to 24 hours in the refrigerator. Drain, reserving 1/2 cup of cola, pat meat dry.
Preheat the oven to 350 degrees F.
Melt the butter in a large skillet over medium-high heat. Add the beef, potatoes, carrots, celery, and onion and saute until browned, about 10 minutes. Add the mushrooms and cook for 5 minutes. Lower the heat to medium, add the flour and cook, stirring constantly, for 2 minutes. Stir in the broth, reserved cola, peas, and brown sugar. Season with salt and pepper, to taste. Let simmer for 15 minutes.
Divide the beef mixture among 4 (20-ounce) casserole dishes or oven-proof bowls. Roll 2 rounds of biscuit dough out to fit over the top of each dish and brush with the beaten egg. Bake until the filling is bubbling and the biscuits are nicely browned, about 30 minutes (cover loosely with aluminum foil if biscuits brown too quickly). Let cool slightly before serving.
Copyright 2002, Janette Barber and Christina Deyo, all rights reserved