In a large mixing bowl, combine 1 1/2 cups of the flour and the yeast; set aside. In a medium saucepan, heat and stir milk, potato, 1/3 cup butter, granulated sugar, and salt just until slightly warm and butter almost melts, add to flour mixture along with eggs. Beat with a mixer on low for 30 seconds. Scraping sides of the bowl constantly. Beat on high for 3 minutes. Stir in as much of the remaining flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough remaining flour to make a moderately soft dough that is smooth and elastic, about 5 minutes. Shape dough into a ball. Place in a lightly greased bowl. Cover; let rise in a warm place until the dough doubles in size, about 45- 60 minutes.
Punch dough down. Turn out onto a lightly floured surface. Cover and let rest for 10 minutes. In the meantime, lightly grease a 13x9-inch baking pan. In a medium bowl stir together 3/4 cup brown sugar, 1/2 cup melted butter, and corn syrup. Spread mixture in prepared pan. For the filling, in a small bowl stir together apples, remaining 3/4 cup brown sugar, and the apple pie spice.
Roll dough into an 18x12-inch rectangle. Spread the 1/4 cup softened butter over dough; sprinkle with apple filling, leaving 1 inch unfilled along one of the long sides. Roll up rectangle, starting from the filled long side. Pinch dough to seal the seam.
Slice rolled rectangle into 12 equal pieces. Arrange over melted butter-brown sugar mixture in pan. Cover and let rise in a warm place until nearly double in size, about 30 minutes.
Preheat oven to 375 degrees F. Bake rolls 25- 30 minutes or until golden. Immediately invert rolls onto a serving platter and spoon any caramel remaining in pan onto rolls. Serve warm.
