Activate the yeast: In the bowl of a stand mixer, whisk warm milk, pumpkin purée, sugar, and yeast. Let sit 5–10 minutes until foamy.
Make the dough: Add melted butter, eggs, flour, salt, cinnamon, and nutmeg. Using the dough hook, knead on medium speed for 6–8 minutes until smooth and slightly tacky.
First rise: Transfer dough to a greased bowl, cover, and let rise in a warm spot for 1–1 ½ hours, or until doubled in size.
Prepare the filling: In a small bowl, mix softened butter, brown sugar, cinnamon, and ginger until well combined.
Shape the rolls: Punch down the dough and roll into a 12×18-inch rectangle. Spread filling evenly over the surface. Roll up tightly from the long side, then cut into 12 even pieces. Arrange in a greased 9×13-inch pan.
Second rise: Cover and let rise for 45–60 minutes, until puffy. Meanwhile, preheat oven to 350°F (175°C).
Bake: Bake for 30 minutes, until lightly golden on top.
Make the frosting: Beat cream cheese and butter until smooth. Add powdered sugar, vanilla, and salt; mix until creamy, adding milk if needed for a spreadable consistency.
Frost and serve: Spread frosting over warm rolls and serve immediately for gooey goodness.