In a large bowl whisk together the warm milk, active sourdough starter, pumpkin puree, granulated sugar, melted butter, and egg until smooth.
Add the flour and salt to the wet mixture and stir with a spatula until a shaggy dough forms and no dry flour remains.
Transfer the dough to a lightly floured surface and knead for 8–10 minutes (or mix in a stand mixer on medium-low for 6–8 minutes) until smooth, elastic, and slightly tacky.
Shape the dough into a ball, place it in a lightly greased bowl, cover with plastic wrap or a towel, and let rise at room temperature for 4–6 hours until puffy and roughly doubled.
While the dough rises, make the filling by stirring together the softened butter, brown sugar, cinnamon, nutmeg, ginger, cloves, and salt in a small bowl until you have a spreadable paste.
Once the dough has risen, gently deflate it and turn it out onto a lightly floured surface.
Roll the dough into a rectangle about 12x18 inches (30x45 cm), keeping the thickness even.
Spread the cinnamon filling evenly over the dough, leaving a 1/2-inch (1.25 cm) border along one long edge.
Starting from the long edge opposite the clean border, roll the dough tightly into a log, pinching the seam to seal.
Use a sharp knife or bench scraper to cut the log into 12 equal pieces.
Arrange the rolls cut-side up in a greased 9x13-inch baking pan, leaving a little space between each roll.
Cover the pan tightly with plastic wrap or a towel and refrigerate for 4–8 hours or overnight for a slow proof.
In the morning, remove the pan from the fridge and let the rolls come to room temperature and proof for 45–90 minutes, or until noticeably puffy and touching.
About 20 minutes before baking, preheat the oven to 350°F (175°C).
Bake the rolls for 22–27 minutes, or until lightly golden on top and the center of a middle roll reaches about 190°F (88°C).
While the rolls bake, make the frosting by beating together the cream cheese and softened butter until smooth and creamy.
Add the powdered sugar, vanilla, salt, and 1 tablespoon of milk or cream, then beat until fluffy, adding more milk as needed for a spreadable consistency.
Let the rolls cool for 10–15 minutes, then spread the cream cheese frosting generously over the warm rolls.
Serve the rolls warm, and store any leftovers covered at room temperature for 1 day or refrigerated for up to 3 days.