Preheat the oven to 375°F (190°C). Lightly butter a cast-iron skillet or baking dish.
Combine the brown sugar and pumpkin pie spice in a small bowl.
Add the salt, baking powder, and baking soda to the rested dough. Work them into the dough thoroughly with your hands so they are evenly distributed.
Transfer the dough to a lightly floured work surface. Roll it into a rectangle, adding a little extra flour to the surface or rolling pin whenever necessary to prevent sticking.
Brush the melted butter evenly over the dough. Sprinkle the pumpkin-spiced brown sugar over the butter, leaving approximately a 1-inch border around the edges.
Starting from one end, roll the dough into a log. Use a sharp knife or bench scraper to divide it into pieces approximately 1–1½ inches thick.
Arrange the Sourdough Pumpkin Cinnamon Rolls in the prepared baking dish, leaving a small amount of space between them.
Bake for 35–40 minutes, or until the rolls are cooked through and beautifully golden brown.
Meanwhile, whisk the powdered sugar, melted butter, alcohol-free vanilla flavoring, and milk until smooth. Add a little additional milk if you prefer a thinner glaze.
Spread or drizzle the glaze over the warm cinnamon rolls before serving.