Sift flour, sugar, baking powder, cream of tartar, and salt into a bowl. Use a pastry cutter to chop vegetable shortening into the flour mixture until it resembles coarse crumbs. Whisk egg and milk together in a separate bowl and slowly add milk mixture to flour mixture, stirring as you pour, until dough is moistened and well-mixed. Drop dough by spoonfuls onto an ungreased baking sheet.
Bake in the preheated oven until biscuits have risen and are golden brown, 10 to 12 minutes.