In large bowl, mix 1 cup of the flour, the yeast, brown sugar, salt and pumpkin pie spice; set aside.
In 1-quart saucepan, heat water, pumpkin and 3 tablespoons of the butter over medium heat to 120°F to 130°F, stirring occasionally. Add pumpkin mixture and egg to flour mixture. Beat with electric mixer on medium speed 3 minutes, scraping bowl occasionally. By hand, stir in just enough of the remaining 2 1/2 to 3 cups flour to make a soft dough that leaves sides of bowl. Place dough on floured surface. Knead 3 to 5 minutes or until dough is smooth and springy.
Place dough in large bowl greased with shortening, turning dough to grease all sides. Cover and let rise in warm place about 1 hour or until double in size.
Place dough on lightly floured surface. Knead a few times. Shape dough into a ball, then flatten. Roll into 15-inch circle. Spread with remaining 1 tablespoon butter. Cut into 16 wedges. Roll up each wedge, starting at wide end. On ungreased cookie sheet, place rolls with points underneath and curve slightly. Cover and let rise in warm place 20 to 30 minutes or until double in size.
Heat oven to 400°F. Bake uncovered 12 to 15 minutes or until golden brown. Serve warm.