In a large bowl, combine the flour, salt, basil and thyme; cut in shortening and butter until crumbly. Add water, 1 tablespoon at a time, tossing lightly with a fork until mixture forms a ball. Cover and chill for at least 30 minutes.
Meanwhile, in a skillet, brown beef in 1 tablespoon oil until no longer pink; sprinkle with salt and pepper. Remove with a slotted spoon; set aside. Add remaining oil to skillet; gradually stir in flour until smooth. Cook and stir over medium heat for about 2-3 minutes or until lightly browned. Gradually add water; whisk until smooth.
Return beef to skillet. Reduce heat; cover and simmer for 20 minutes. Add the potatoes, carrot and onion. Cover and simmer for 25 minutes or until tender. Remove from the heat; cool.
Divide pastry into four equal portions. On a lightly floured surface, roll out one portion into a 9-in. circle. Mound 3/4 cup filling on half of circle. Moisten edges with water; fold dough over filling and press edges with a fork to seal.
Place on an ungreased baking sheet. Repeat with remaining pastry and filling. Cut three slits in top of each; brush with egg.
Bake at 400° for 25-30 minutes or until golden brown.