Place butter in a 3-quart rectangular baking dish and set dish in a 350 degree F oven to melt butter, allowing 5 to 8 minutes. (Be sure to watch carefully to avoid overbrowning the butter). Remove dish from oven; set aside.

For syrup: In a medium saucepan, combine the 2 cups sugar and water. Cook and stir over medium heat until sugar is dissolved; bring to boiling and boil, uncovered, for 5 minutes. Remove saucepan from heat; stir in vanilla. Cover and keep warm.

For dough: Place 2 cups flour in a large mixing bowl. Use a pastry blender to cut shortening into flour until pieces are the size of small peas. Make a well in center. Add milk all at once. Stir just until moistened. Knead dough on a lightly floured 14x12-inch piece of waxed paper 10 to 12 strokes or until nearly smooth. Lightly sprinkle dough with flour. Cover with another 14x12-inch piece of waxed paper. Roll out dough to a 12x10-inch rectangle (about 1/4 inch thick).

For filling: In another large bowl, combine peaches, cinnamon and nutmeg. Spoon 3 cups of the peaches on top of the melted butter in the baking dish, spreading evenly to form a single layer. Spoon the remaining 2 cups peaches evenly over the dough rectangle. Roll dough into a spiral, starting from a long side. Pinch seam to seal. Cut into twelve 1-inch-thick pieces. Arrange rolls, cut sides down, over the peaches in the dish. Pour syrup carefully around rolls. (This will look like too much syrup, but the rolls will absorb the liquid during baking.)

Bake in the 350 degree F oven for 45 to 50 minutes or until golden. Cool on a wire rack 30 minutes. Serve warm in dishes with a scoop of vanilla ice cream. Makes 12 servings.