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Prepare your yeast mixture by mixing together the milk, butter, and granulated in a small bowl. Microwave until the liquid reaches at least 115-120F degrees. You do not want it to be warmer than 120F degrees because the yeast will not activate. Mix in the yeast and pour into a large mixing bowl. Allow the mixture to set for 10 minutes to allow the yeast to activate and foam.
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Using a paddle attachment on a stand or a handheld mixer, mix in the egg and pumpkin puree.
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Using the dough hook, slowly add in the flour, ½ cup at a time, then add in the pumpkin pie spice. Knead the dough for 10 minutes on low. Place a kitchen towel on top of the mixing bowl and allow the dough to rise for 60-90 minutes, or until the dough has doubled in size. Once the dough has doubled, punch the dough, and allow it to rise for another 30 minutes.
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Flour your counter top and place the dough on top, cut into 9-10 pieces. I cut mine into 8 pieces and the rolls are pretty huge. Roll each piece of dough into a roll form, making sure to tuck the edges underneath the roll.
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Place the rolls into a 9x9 baking dish, cover them with the kitchen towel again and allow them to rise for another 60-90 minutes, or until they have doubled in size.
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Preheat oven to 375 degrees and allow the rolls to bake for 15-17 minutes, or until the tops have browned.