Combine first 8 ingredients in a Dutch oven. Cover with water and bring to a boil; cover, reduce heat, and simmer 1 hour or until tender. Remove chicken from broth; strain broth, reserving 1 1/2 cups. Cool chicken, remove from bone, chop.

Saute celery, onion, carrot and potato in butter until crisp-tender. Add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add reserved broth and half and half. Cook over medium heat, stirring constantly until thickened and bubbly. Stir in salt, pepper and the chopped chicken.

Combine flour and salt; cut in shortening with pastry blender until mixture resembles coarse meal. Combine egg, water, and vinegar; sprinkle evenly over surface and stir with a fork until dry ingredients are moistened.

Shape into a ball. Roll half of pastry to 1/8-inch thickness on lightly floured surface. Fit into a 9 1/2 inch deep-dish pie plate. Spoon chicken mixture into prepared pastry.

Roll remaining pastry to 1/8-inch thickness and place over chicken filling. Trim, seal, and flute edges. Cut slits in top to allow steam to escape. Combine egg and milk; blend well. Brush over pastry. Bake at 400 degrees for 30 minutes or until golden brown. Yield: 6 servings.