Combine chicken, potatoes, onion, carrots, 1/4 cup water, salt, and pepper. Divide pastry into eight equal portions; roll out each portion into a 6-1/2" circle. Spoon 1/2 cup meat mixture on the center of each circle. Beat together egg and water; brush edges of circles with beaten egg, making a 3/4" border. Gently lift sides of circle to meet top. Using fingers, firmly press edges together; flute edges. Combine 3 cups flour and salt in a bowl; cut in shortening with pastry blender until mixture resembles coarse meal. Sprinkle cold water (1 T. at a time) evenly over surface; stir with a fork until all dry ingredients are moistened.

Place pies on ungreased baking sheets; bake at 350 F. for 1 hour or until lightly browned.