Prep the yeast. In a small saucepan, heat the milk and 1 tablespoon sugar until just warmed, 90 to 105°F. Do not heat over 110°F or you will risk killing the yeast. Once warmed, remove from heat and stir in the yeast. Let it sit for 5 minutes until the yeast is bubbly. Whisk in the eggs and vanilla.
Mix the dry ingredients. In the mixing bowl of a standing mixer with the paddle attachment, combine the remaining sugar (⅓ cup minus 1 tablespoon), flour, and salt.
Make the dough. With the mixer on the lowest speed, slowly add the milk mixture until combined. Add the butter 1 piece at a time beating until incorporated before adding the next piece. Increase the speed to medium-high for 5 minutes. The dough will be silky, smooth and have pulled away from the sides of the bowl.
Rise. Transfer the dough to a lightly greased bowl and cover with plastic wrap or a kitchen towel. Let rise for 1 to 2 hours in a warm spot or until double in size. While the dough rises, grease a 9×13 baking dish with butter and set aside.
Roll out the dough. Once the dough has doubled in size, tip the dough out onto a well-floured surface. Roll out into an 18″ x 12″ large rectangle.
Spread the filling. Brush the melted butter over the dough, saving about a tablespoon for the formed rolls. Sprinkle the granulated sugar and brown sugar evenly over the buttered dough. Mix together the pumpkin puree and pumpkin pie spice. Using the backside of a spoon gently spread the purée over the dough. Dot with pieces of soft mascarpone over the dough.
Form the rolls. Tightly roll the dough from the long side. Using a string or a sharp serrated knife, cut the dough log into 1 ½″ rounds. Brush the tops with melted butter, cover with a kitchen towel, and let rise for 30 to 45 minutes in a warm place.
Bake. Preheat the oven to 350°F. Bake the buns for 45 to 50 minutes – cover the buns with aluminum foil for the last 15 minutes of baking if they’ve started to brown by the 30-minute mark.
Make the frosting. In a standing mixer with the whisk attachment, whisk the mascarpone cheese until light and fluffy. Add the confectioner's sugar to the mascarpone and with the mixer on low speed, pour in the heavy cream and vanilla slowly increasing the speed until the frosting comes together. Add the pumpkin puree and pumpkin pie spice mixing until combined. If the frosting separates slightly, it will smooth out when spread over the hot buns.
Frost the cinnamon rolls. Pour the frosting over the freshly baked buns and serve immediately. If serving at a later time, reheat the buns in the baking dish then pour the frosting over just before serving.