In a small bowl, combine warm milk and yeast. Let it sit for about 5 minutes until frothy.
In a large mixing bowl, whisk together pumpkin puree, sugar, melted butter, egg, and vanilla extract. Add the yeast mixture and stir to combine.
In another bowl, mix flour, salt, cinnamon, nutmeg, and ginger. Gradually add the dry ingredients to the wet mixture, stirring until a dough forms.
Knead the dough on a floured surface for about 5-7 minutes until smooth.
Place the dough in a greased bowl, cover with a kitchen towel, and let it rise in a warm place for about 1 hour or until doubled in size.
Once risen, roll the dough out on a floured surface into a rectangle (about 12x18 inches).
Spread softened butter over the dough, then sprinkle brown sugar and cinnamon evenly on top.
Roll the dough tightly from the long side and cut into 12 equal pieces.
Place the rolls in a greased 9x13 inch baking dish. Cover with a towel and let them rise for another 30 minutes.
Preheat the oven to 350°F. Bake the rolls for 25-30 minutes or until golden brown.
While the rolls are baking, prepare the frosting by whisking together powdered sugar, brewed coffee, maple syrup, softened butter, and vanilla extract until smooth.
Once the rolls are out of the oven, let them cool for about 10 minutes before drizzling with the coffee maple frosting.