Stir the yeast and a pinch of sugar into the warm milk and let sit 5 minutes until foamy.
Blot the pumpkin puree with a paper towel, then whisk it with the remaining sugar, melted butter, egg, pumpkin pie spice, and salt. Stir in the yeast mixture.
Add the flour a cup at a time until a soft dough forms. Knead 6 to 8 minutes until smooth and springy.
Place in a greased bowl, cover, and let rise in a warm spot about 1 hour, until doubled.
Fill and Shape
Stir the softened butter, brown sugar, cinnamon, and pumpkin pie spice into a soft paste.
Roll the dough into a 16 by 12-inch rectangle. Spread the filling to the edges, roll up snugly from the long side, and cut into 12 rolls (a length of dental floss cuts them cleanly).
Arrange the rolls in a greased 9 by 13-inch baking dish. Cover and let rise 30 to 45 minutes, until puffy and touching.
Bake and Ice
Bake at 350F for 25 to 30 minutes, until golden and set in the center. Cool 10 minutes.
Beat the cream cheese and butter until smooth, then beat in the powdered sugar, maple syrup, and vanilla. Spread over the warm rolls.
Notes
Blot the pumpkin: Patting out extra moisture keeps the dough from turning sticky.
Overnight: Shape the rolls, cover, and refrigerate overnight. In the morning let them sit out 30 to 45 minutes, then bake.