In a large mixing bowl add warm water, yeast and sugar, stirring together. Mix in egg and pumpkin.
Add 2 cups of flour and salt, mixing with a dough whisk or wooden spoon. Continue to mix, adding 1/2 cup of flour. until dough forms a ball and starts to pull away from sides of bowl.
Pour dough onto a floured surface and knead for 3- 5 minutes, adding flour as needed until dough is soft and smooth but no longer sticky. Allow dough to rest about 5 minutes.
Use a rolling pin to roll out to a rectangle shape about 12 x 18 inches.
Spread softened butter in a thin layer over the dough. Sprinkle with brown sugar and pumpkin pie spice.
Starting from the long side, carefully roll up dough,.
Butter a 9 x 13 inch baking dish.
Using a serrated knife cut into 1 1/2 inch slices and place into the prepared baking pan.
Cover and rise 30 - 45 minutes until doubled in size.
Preheat oven to 350°.
Bake for 20 - 22 minutes or until rolls are cooked through and lightly browned.
Remove to a wire rack to cool.
Make the icing. Add softened cream cheese, softened butter, milk, vanilla and powdered sugar to a bowl. Mix with an electric mixer or wooden spoon until smooth.
Pour over warm pumpkin cinnamon rolls, gently spreading to cover the rolls.
Serve warm or at room temperature.