Combine flour, granulated sugar, pumpkin pie spice, instant yeast and salt in the bowl of a stand mixer*. Mix until combined. Add the pumpkin purée, sourdough discard, milk and egg. With the mixer on low speed, slowly pour in the melted butter, mixing until a shaggy dough forms.
Transfer the dough to a smooth work surface and knead until the dough is smooth and soft, about 3-4 minutes. The dough should not be sticky. If needed, add more flour or milk (1 Tablespoon at a time) to reach your desired consistency. Transfer the dough to a large, greased bowl. Cover with plastic wrap and let rise for 1 ½-2 hours, or until doubled in size.
Transfer the dough to a lightly floured work surface. Use a rolling pin to roll the dough into a 12x18-inch rectangle**. Spread the dough with softened butter and sprinkle with brown sugar and cinnamon, using your hand to spread the sugar evenly.
From the long end of the rectangle, roll the dough into a log, pressing the seam along the edge of the dough to close. Use unflavored dental floss or a sharp knife to cut the log into 12 approximately equal-sized pieces. If using dental floss, wrap the floss around the log, then cross the ends over each other and pull, pinching to cut.
Transfer the cut rolls to a greased 9x13-inch baking pan, face up. There should be room for the rolls to expand in the pan. Cover with plastic wrap and let rise for 30-45 minutes until the rolls have expanded to touch each other.
Preheat the oven to 350°F.
Bake the rolls for 30-35 minutes, until golden brown. Remove from the oven and let cool.
While the rolls cool, make the frosting. In a medium bowl, whip the softened butter until light and fluffy, about 2-3 minutes. Add vanilla extract and a pinch of salt. Add the confectioner's sugar and 1 Tablespoon milk, mixing until smooth. Add additional milk to reach your desired consistency, if needed. Spread the frosting on the rolls and serve.
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Notes
*You can make this recipe without a stand mixer. Mix the dough in a large bowl with a wooden spoon, then knead by hand. **If you are doubling or tripling this recipe, divide the dough into 2 (for doubling) or 3 (for tripling) portions. Then divide each portion into individual rolls and continue with the directions as written. To make this recipe with active sourdough starter, omit the yeast and replace the sourdough discard with active starter (200g). You will also need to increase the rise times.