Add the warm milk, sugar, pumpkin puree, active sourdough starter, whole egg, egg yolk, vanilla extract, salt, bread flour, and softened butter to the bowl of a stand mixer.
Using the dough hook, mix on low speed until the ingredients come together into a rough dough. Cover and let the dough rest for about 30 minutes.
Knead on low speed for another 2–3 minutes, or until the dough becomes smooth, elastic, and well combined.
Cover the bowl and allow the dough to bulk ferment at room temperature. Look for approximately a 30% increase in volume rather than waiting for the dough to double. It should feel noticeably softer and slightly aerated.
Meanwhile, prepare the filling by combining the melted butter, brown sugar, and cinnamon or pumpkin pie spice. Mix until it has a wet, sandy consistency.
Turn the fermented dough onto a clean work surface. Gently roll it into a rectangle measuring approximately 20 x 12 inches (50 x 30 cm) and about 1/4 inch (1/2 cm) thick.
Spread the cinnamon filling evenly over the dough, leaving about 2 inches (5 cm) uncovered along one long edge.
Starting from the opposite long edge, tightly roll the dough into a log. Finish with the seam underneath.
Cut the log into 12 approximately equal rolls. Arrange the Sourdough Pumpkin Cinnamon Rolls in a 9 x 13-inch baking dish, positioning the seams inward to help prevent them from unraveling.
Cover the baking dish and leave the rolls to rise until noticeably soft and puffy. They should expand enough that they are beginning to touch one another.
Preheat the oven to 350°F (180°C).
For extra-rich, soft pumpkin cinnamon rolls, gently pour the heavy whipping cream over the risen rolls immediately before baking.
Bake for 30–35 minutes, or until the rolls are puffed and golden. Allow them to cool in the baking dish for approximately 30 minutes.
To make the icing, beat the cream cheese, softened butter, powdered sugar, vanilla, and salt with a whisk attachment until smooth, thick, and glossy.
Spread the vanilla cream cheese icing generously over the slightly warm rolls and serve.