Combine warm milk (105–115°F / 40–46°C), 1 tablespoon of the sugar, and the yeast in the bowl of a stand mixer; stir gently and let sit for 5–10 minutes until foamy and bubbly. If no foam forms, the yeast is inactive — discard and start over with fresh yeast.
Add the pumpkin puree, melted butter, egg, vanilla, salt, pumpkin pie spice, and remaining sugar to the yeast mixture; mix on low to combine, then add 3½ cups flour and knead with the dough hook on medium for 6–8 minutes until smooth and slightly tacky, adding flour one tablespoon at a time only if the dough sticks to the bowl.
Shape the dough into a ball, place in a lightly greased bowl, cover with plastic wrap, and let rise in a warm spot for 1 to 1½ hours until doubled in size.
Stir together the brown sugar, cinnamon, pumpkin pie spice, and nutmeg in a small bowl; set aside and ensure your butter is fully softened to room temperature.
Punch down the risen dough and roll it out on a lightly floured surface into a 14x18-inch rectangle about ¼ inch thick; if it springs back, let it rest 5 minutes and try again.
Spread the softened butter evenly over the dough leaving a ½-inch bare border along the far long edge, then sprinkle the brown sugar spice mixture over the butter and press it in lightly with your hands.
Tightly roll the dough from the long edge closest to you into a log and pinch the seam to seal, then cut into 12 equal pieces (about 1.5 inches thick) using a sharp knife or unflavored dental floss.
Arrange the rolls cut-side up in a greased 9x13-inch baking dish, cover loosely, and let rise in a warm spot for 30–45 minutes until puffed and touching. (Overnight option: refrigerate covered, then rest at room temperature 30–45 minutes before baking.)
Preheat oven to 350°F (175°C) and bake for 23–27 minutes until the tops are golden and the centers are cooked through, tenting loosely with foil if browning too quickly; cool in the pan for at least 10 minutes before frosting.
Beat the softened cream cheese until smooth, mix in the sifted powdered sugar, vanilla, and salt, then add milk one tablespoon at a time until your desired spreading or drizzling consistency is reached.
Spread the cream cheese frosting generously over the warm rolls and serve immediately while the frosting melts into every swirl.