Mix 3 cups of the flour with the rest of the dry ingredients together in large bowl and create a well in the center.
Add the egg, butter, pumpkin, and water to the dry ingredients. Mix until combined.
Add the remaining cup of flour and mix until combined.
Cover and let dough rest for 10 minutes.
On a floured surface, turn out the dough and knead until it comes together and is smooth, about 6-8 minutes by hand or 4 minutes in a stand mixer.
To Assemble the Cinnamon Rolls
Combine the pumpkin, butter, brown sugar, cinnamon, and salt.
Roll out dough into a rectangle, about 1/4 inch thick.
Spread the pumpkin mixture out on top of the dough.
Roll the rectangle up tightly (you will be rolling the long side) so that you have a long rolled log.
Cut the log into 12-15 even rolls. This is most easily done by dividing the dough into 4 pieces and then each of the 4 pieces into three pieces (for 12 cinnamon rolls) or dividing the dough into 3 pieces and then each of the 3 pieces into 5 pieces (for 15 cinnamon rolls).
For rolls with soft sides, place them 1/2 inch apart on a sheet tray, in a pie plate, or in a cast iron skillet. For rolls that do not touch, place 1 1/2 inches apart. Be sure to grease whatever you use to bake them in.
Cover rolls and let rest for 15 minutes.
Uncover and bake at 350 degrees F for about 20 minutes, the tops will just barely start to turn golden and the center cinnamon roll will be cooked through.
For the Icing
Combine cream cheese and butter until smooth.
Add vanilla extract and mix well.
Add in half of the powdered sugar and the milk, mix until smooth.
Add in the rest of the powdered sugar until your desired consistency is reached, I like the icing to be thick but pourable.
Pour icing over rolls as soon as they are removed from the oven.
The exact amount of water you will need for this recipe (or any baking recipe) can vary quite a bit. This is due to altitude, humidity, and weather. I live in an arid high-desert mountain climate, so often my readers need to less water if they live at a lower elevation with more humidity. If too much water is added, simply add more flour gradually until the proper consistency is reached.