Activate the yeast: In the bowl of a stand mixer, combine warm milk, 1 tsp of the granulated sugar, and yeast. Stir gently and let sit 5–10 minutes until foamy.
Mix the dough: Add the remaining sugar, egg, melted butter, pumpkin puree, salt, and pumpkin pie spice to the yeast mixture. Mix on low until combined.
Add flour and knead: Gradually add 3 ½ cups (440 g) flour, one cup at a time, mixing until a soft dough forms. Increase speed to medium-low and knead 5–7 minutes, adding more flour as needed until the dough is smooth and slightly tacky but not sticky.
First rise: Transfer dough to a lightly greased bowl, cover with plastic wrap or a damp cloth, and let rise in a warm place 1–1½ hours, or until doubled in size.
Prepare the pan: Grease a 9×13-inch (23×33 cm) baking pan with butter or non-stick spray.
Roll out the dough: Punch down the dough and turn it onto a lightly floured surface. Roll into a 12×16-inch (30×40 cm) rectangle.
Add filling: Spread softened butter evenly over the dough. In a small bowl, mix brown sugar, cinnamon, and pumpkin pie spice; sprinkle evenly over the butter. If using, spread the optional pumpkin-sugar mixture in a thin layer.
Shape the rolls: Starting from a long side, roll the dough tightly into a log. Pinch the seam to seal. Using a sharp knife or dental floss, cut the log into 12 equal pieces.
Second rise: Place rolls cut-side up in the prepared pan, cover, and let rise 30–45 minutes until puffy and nearly doubled. Meanwhile, preheat oven to 350°F (175°C).
Bake: Bake rolls for 20–25 minutes, until light golden brown and the centers look set. If you have a thermometer, the internal temperature should read about 190–195°F (87–90°C).
Make the frosting: While rolls bake, beat cream cheese and butter until smooth. Add powdered sugar, vanilla, and salt; mix until creamy. Add a little milk or cream if needed to reach a spreadable consistency.
Frost and serve: Let rolls cool 5–10 minutes, then spread or drizzle the cream cheese frosting over the warm rolls. Serve warm or at room temperature.