Warm milk in a small pot till around 105-110°F.
Pour warm milk into stand mixer bowl if using a mixer or a large bowl if mixing and kneading by hand.
Add sugar, and sprinkle yeast on top. You can let this rest a couple minutes to make sure your yeast is happy before proceeding.
Stir in egg, pumpkin purée, sourdough starter, and melted butter until mixture is creamy and combined well.
Add flour, salt, and pumpkin pie spice and mix or stir in until sticky dough begins to form.
Add dough hook to mixer and mix on low until a ball is formed, after approximately 10 minutes. Dough ball should be slightly sticky. If kneading by hand, place dough on a well floured surface and knead for about 10 minutes.
Transfer dough ball to a well greased bowl, cover with a tea towel, and allow to rise until doubled in size for about an hour.
After dough has doubled in size move to a well floured surface. Flour your rolling pin as well. Roll dough into a rectangle of approximately 14x16 inches.
Spread soft butter over the dough leaving a margin of about ¼ inch at the far end (short end).
Mix brown sugar and cinnamon in a small bowl and then sprinkle over the rolled out dough, gently rubbing it into the butter.
From the shorter side, roll dough as tightly as possible to the end with the ¼ margin and place seam side down.
You can cut off the ends, so rolled dough is one nice 12-inch log. Make score lines so you know where to cut evenly.
Ready your 9x13" pan by lining with parchment paper and greasing.
Using a sharp knife, cut the dough into 1-inch rolls and snugly place rolls in a 3x4 grid down the length of the pan.
Cover with plastic wrap or tea towel and let raise for half an hour, or put in fridge overnight and bake the next day if you wish.
If you are baking right away, remove towel and preheat oven to 350°F and place in oven when 30 minute rise is up.
If you are baking the next day, take rolls out of the fridge and let warm for at least half an hour. Take off plastic wrap. Pre-heat your oven to 350°F while second rise is happening.
Bake for 20 minutes until rolls are slightly golden brown on the edges and still soft in the middle. Cool for 15 minutes.
For the cream cheese frosting, mix soft cream cheese, butter, maple syrup, icing sugar and salt in a small bowl until smooth and creamy.
Spread over rolls and serve.