In the bowl of a stand mixer, combine the warm milk, yeast, and 1 tablespoon of the granulated sugar. Let sit for 5-10 minutes until foamy.

Add the remaining sugar, pumpkin puree, melted butter, and egg. Whisk until smooth.

Add the flour and salt. Attach the dough hook and knead on low speed for 5-7 minutes until the dough is smooth and slightly tacky, but not overly sticky. Place in a greased bowl, cover with a warm towel, and let rise for 1 hour until doubled in size.

Punch the dough down and turn it out onto a lightly floured surface. Roll into a 12x18 inch rectangle.

In a medium bowl, mix the softened butter, brown sugar, pumpkin puree, cinnamon, and pumpkin pie spice until it forms a thick paste. Spread this filling evenly over the dough, leaving a small border at the edges.

Starting from the long edge, tightly roll the dough into a log. Cut into 12 even slices using a serrated knife or unflavored dental floss.

Place the rolls into a greased 9x13 inch baking dish. Cover and let rise for 30-45 minutes until puffed up. Preheat oven to 350°F (175°C).

Bake for 22-26 minutes until the rolls are golden brown on top and cooked through in the center.

While the rolls bake, beat the cream cheese and butter until smooth. Add the powdered sugar, vanilla, and maple syrup, beating until fluffy.

Let the rolls cool for 5 minutes before generously spreading the frosting over the top so it melts down into the spirals.