Preheat oven to 450°F. In a medium saucepan combine potatoes and 1/4 teaspoon salt; add enough water to cover. Bring to boiling; reduce heat. Cover and simmer for 10 to 12 minutes or just until tender. Add the parsnips to the potatoes during the last 4 minutes of cooking. Drain; cool slightly.
Meanwhile, in a very large skillet cook chicken in hot oil over medium heat about 5 minutes or until brown, stirring occasionally. Remove chicken from skillet; set aside. Add onion, celery, and mushrooms to skillet. Cook and stir about 10 minutes or until vegetables are tender. Return chicken to skillet. Stir in barbecue sauce and cooked potatoes and parsnips. Heat through. Divide mixture evenly among six 10-ounce ramekins or custard cups. Set aside
Unroll pie crust. Press or roll pie crust into a 12-inch circle. Cut crust into 1/2-inch-wide strips. Cut long center strips in half. Arrange half of the strips 1-inch apart on filling (about 3 strips per ramekin). Give each ramekin a quarter turn; arrange remaining strips perpendicular to the first half of the strips (about 3 strips per ramekin).
In a small bowl whisk together the egg and the water. Brush egg mixture on lattice crusts.
Bake, uncovered, for 15 to 20 minutes or until the filling is bubbly and crust is golden brown. Let stand for 5 minutes before serving.