Combine all filling ingredients and cook until thick and caramelized (approximately 35 min). Dough: Combine flour and sugar; cut in shortening with pastry blender. Make well in center and add beer all at once.
Stir with a fork until mixture becomes a soft dough. Shape dough into ball, pinch off small amounts and shape into patties. Allow patties to rest approximately 20 minutes.
Roll out on a lightly floured board and spoon pumpkin mixture onto circle of dough. Fold over and seal edges together by pressing with a fork. Prick top of turnover to allow steam to escape.
Place turnovers on ungreased cookie sheet and bake at 350 for 15-20 minutes or until golden brown.