In a small, heat proof measuring cup, combine the milk and 2 tablespoons of the butter. Microwave on high for about 30 seconds or just until the milk is warm to the touch and the butter begins to melt.
Pour the warm milk and butter into the bowl of a stand mixer fitted with the whisk attachment. Add the granulated sugar, salt, egg, and pumpkin and whisk together on medium speed until well combined, scraping the sides of the bowl with a spatula as needed.
Next, add the bread flour to the pumpkin mixture and sprinkle in the instant yeast. Switch the attachment to the dough hook, and knead the dough for five minutes, using a spatula to scrape the sides as needed. (If you don’t have a stand mixer, you can knead the dough by hand instead, but you will need to add a little extra flour, one tablespoon at a time, as the dough will be very sticky and may be difficult to knead by hand).
Spray a medium sized bowl with cooking spray. Rub a little flour on your hands (the dough will be sticky), and then move the dough from the mixer to the greased bowl. Cover with plastic wrap and place in a warm place to rise (I like to stick mine inside my microwave oven). Let the dough rise for 45-60 minutes or until it has doubled in size.
Meanwhile, let the 2 remaining tablespoons of butter soften at room temperature in a small dish. In a separate small bowl, mix together the brown sugar, cinnamon, nutmeg and set aside. Prepare a 9x13 inch baking pan with cooking spray.
When the dough has risen and doubled in size, sprinkle your clean work surface with a few tablespoons of flour. Punch the dough down and then transfer to your floured work surface. Sprinkle a little more flour on top of the dough, and then knead it for 5 or 6 turns until it forms a smooth ball that is no longer sticky. Using a rolling pin, roll the dough into a 14x10 inch rectangle.
Using a pastry brush or butter knife, spread the soft butter over the dough all the way to the edges.
Next, sprinkle the sugar and spice mixture over the dough, leaving about a ¼ inch space around the edges.
Beginning at the long end, roll the dough up like a jelly roll. Roll the ends under and give them a pinch to seal the edges. Using a sharp knife, slice the roll into twelve pieces.
Place the individual rolls into the prepared pan. Let the rolls rise for 30-45 minutes or until doubled in size. (If the pan will fit in your microwave, great, otherwise choose another warm spot in your kitchen).
When the rolls are close to completing their rising time, heat your oven to 350°. Place the baking pan into the hot oven and bake for 20-25 minutes or until the rolls are golden on top and sound hollow when tapped.