Preheat oven to 400 F / 200 C.
Rub the bottom of a 9"x9" baking dish with 4 Tbsp of softened butter.
In the bowl of a stand mixer fitted with the dough hook, combine warm water, yeast, and sugar. Let stand until creamy, about 5 minutes.
Meanwhile, dust a kitchen counter lightly with flour.
To the yeast mixture, add 2 Tbsp butter, salt, pumpkin pie spice, and 2 cups of flour. Mix on low until the ingredients come together.
Continuing to mix on low, add 1 more cup of flour and mix for 1 minute. Add the remaining 1 cup flour. Mix on low until all the flour is mixed in. Increase mixer speed to medium. Mix for 5 minutes, until the dough is smooth and elastic and begins to make a slapping sound as it hits the sides of the bowl.
Remove the bowl from the mixer and let the dough rest in the bowl (preferably near the warm oven), until the surface is smooth, about 5 minutes.
While dough rests, make filling by combining pumpkin, brown sugar, and cinnamon.
Turn the dough out on a floured surface and roll it a 9"x18" rectangle with a long side facing you. (Note: Don't worry about being precise; use the base of your 9"x9" baking dish as a general guide.) Use your hands to rub 4 Tbsp of butter over the surface of the dough.
Spread pumpkin filling over the surface of the dough. .
Starting with the side closest to you, roll the dough into a tube. Use a serrated knife to cut 9 even slices. Place the slices in the prepared baking dish and let rise in a warm place (preferably near the warm oven) until the edges of the rolls are touching, 20 minutes.
Bake until golden brown on top, 20-25 minutes.
While the rolls are baking, use a hand mixer to beat 3 Tbsp of butter and cream cheese until light and fluffy. Add powdered sugar, vanilla, and 3 Tbsp milk. Continue mixing until icing is smooth. Add more milk a bit at a time, if needed, until icing is shiny and easy to pour.
When the rolls come out of the oven, let them cool for 5 minutes before topping with the icing.
Serve immediately (of course...who could wait!).