In a small bowl, stir together the milk, melted butter, pumpkin puree, and sugar. The mixture will be thick. Microwave until the mixture is warm (about 110 degrees on an instant read thermometer). Stir in the active dry yeast, and set aside to bloom.
In a large bowl, stir together the egg and the vanilla extract. Once the yeast mixture has bloomed, it should look a little puffy near the yeast granules, stir the yeast mixture into the egg and vanilla until incorporated.
Add in the salt, cinnamon, nutmeg, allspice, and 3 cups of flour. Stir together with a sturdy spatula or a wooden spoon until a shaggy dough forms. Turn the dough out onto a floured surface and gently start to knead. You will likely need to add a bit more flour while kneading. You are looking for the dough to still be slightly tacky and soft, but strong enough to hold in a ball shape. It should only take about 5 minutes of kneading in additional flour.
Return the dough to the bowl and cover with plastic for the first rise. You want the dough to grow by just under double in size, as you can see from the photos above. For me, this only takes about 20-30 minutes.
Return the dough to your floured surface, and begin to pat out into a rectangle. Then use a rolling pin to roll the dough to a rectangle about 10x16 inches. It doesn't need to be exact.
Spread the softened butter evenly over the dough, all the way to the edges. It will be a very thin layer of butter. In a small bowl, stir together the brown sugar and cinnamon, and then evenly sprinkle the mixture over the butter, again all the way to the edges. With your rolling pin, gentle press the mixture into the dough. Starting on the longer edge, roll your dough into a spiraled log.
Cut of the edges of the dough, and then cut the dough into 8 even rolls. I use a very sharp knife and a sawing motion here, cutting the dough in half, and then each of those pieces in half, and repeating again to reach 8 total.
Lightly grease an 8 inch cake pan with cooking spray, and place the rolls evenly into the pan. Cover with plastic for the second rise. Preheat your oven to 375 while the dough rises.
For the second rise, the dough will only get slightly puffed. For me, this takes about 15 minutes, or about as long as it takes for my oven to preheat. Then, remove the plastic and bake the rolls for 25-30 minutes, until just slightly golden on top.
In a small bowl, whisk together the softened cream cheese, powdered sugar, maple extract, salt, and 1/2 tbs. of milk until smooth. Add more milk if needed to reach your desired consistency, though I like to keep my icing a bit thick.
Frost the rolls in the pan while still hot, and serve warm. Enjoy!