Combine the warm water with the yeast and about 1 teaspoon of the granulated sugar. Let sit to bloom four about 5-10 minutes.
Meanwhile, combine the flour, salt, pumpkin spice, and the remaining granulated sugar in the bowl of a standing mixer. When the milk is ready, add that in along with the egg, melted butter, pumpkin, and vanilla. Using the dough hook attachment, start at a slow speed and then increase to medium-high. Do this until a somewhat sticky dough has formed, about 5 minutes.
Add a little flour to a clean work surface and pour the dough out. Work it for just about a minute or so, or until it pulls together into a dough ball. Add it to an oiled bowl. Cover with plastic wrap and a kitchen towel. Let rise in a warm place (I use the oven with the light on) for about 2 hours, or until doubled in size.
While that's rising, make the filling. First pat the pumpkin dry by spreading it on a plate and using a paper towel. Then combine it with the brown sugar, butter, and pumpkin spice. Mix until a paste forms and set aside.
When the dough is ready, remove it and preheat the oven to 375 degrees Fahrenheit. Add a bit of flour to a clean work surface and add the dough. Roll it out to a rectangle about ¼ inch thick. Spread the filling all over. Make your rolls the classic way by rolling the dough lengthwise into a log, using kitchen twine to cut out your rolls. Or you can use a pizza cutter to cut long strips and roll them that way. Either way, add them all to the baking dish, making sure there's a little room for them to rise. Cover with plastic wrap and let them rise for about 45 minutes, or until they're touching slightly. Before baking, melt that tablespoon of butter and brush it on top of the rolls.
Bake in the preheated oven for about 20 minutes, or until they're lightly golden. While baking, make the cream cheese frosting by whipping all of the ingredients together for 5-10 minutes, or until fluffy.
Remove the rolls and add a bit of the frosting while they're still warm.