Divide pastry into 8 equal portions; roll each to 1/4-inch thickness on a lightly floured surface. Trim edges to form a square.
Place 1 peach half, cut side up, in center of each pastry square. Combine sugar and cinnamon; spoon 1 tablespoon of sugar mixture into center of each peach half. Dot with butter. Place remaining peach halves, cut side down, over sugar mixture. Moisten edges of each dumpling with water; bring comers to center, pinching edges to seal. Place dumplings, seam side down, in a lightly greased 13- x 9- x 2-inch baking dish.
Bake at 450° for 15 minutes or until lightly browned. Reduce heat to 300°; pour half of syrup over dumplings. Stir vanilla into remaining syrup. Bake an additional 30 minutes, gradually adding additional syrup so dumplings will not bake dry.