Make the Dough: In a large bowl, whisk together 1 cup of flour, sugar, yeast, cinnamon, nutmeg, and ginger. Set aside.
Heat the Wet Ingredients: In a heatproof bowl or measuring cup, microwave the milk and butter for 30–45 seconds, until warm to the touch (about 100–110°F). Stir in pumpkin purée until smooth. Cool slightly if too hot.
Combine and Knead: Pour the warm pumpkin mixture and ½ beaten egg into the flour mixture. Stir with a wooden spoon until a thick, sticky batter forms. Add another 1 cup of flour and mix until combined, then gradually add ¼ cup more at a time until a soft dough forms. Transfer the dough to a lightly floured surface and knead for 6–8 minutes, until smooth and elastic. If sticky, add up to ¼ cup more flour. The dough should spring back when pressed.
Rest and Prepare Pan: Let the dough rest uncovered for 10 minutes. Meanwhile, butter an 8"x8" baking pan.
Make the Gooey Bottom Layer: In a small bowl, whisk together melted butter, brown sugar, white sugar, cream, vanilla, and salt until smooth. Pour this mixture evenly into the bottom of the prepared baking pan. Set aside.
Shape and Fill the Rolls: Roll the rested dough into a 12” x 7–8” rectangle. Spread the softened butter evenly over the dough, leaving a ½-inch border. In a small bowl, combine the brown sugar, cinnamon, and pumpkin pie spice. Sprinkle evenly over the buttered dough.
Roll and Cut: Roll the dough tightly from the long edge into a log. Press the ends to seal, then cut into 6 equal pieces using unflavored dental floss or a sharp knife.
Assemble and Rise: Place the rolls cut-side down over the gooey mixture in the pan. Cover loosely and let rise in a warm spot for about 1 hour, or until puffy and nearly doubled in size.
Bake: Preheat oven to 350°F (325°F convection). Bake for 25–30 minutes, until golden on top and bubbling at the bottom. Let cool for about 10 minutes before frosting.
Make the Frosting: In a medium bowl, mash the softened butter and cream cheese together until smooth. Add pumpkin purée (if using), powdered sugar, vanilla, and cinnamon. Whisk until light and fluffy.
Frost and Serve: Spread the frosting generously over the warm rolls so it melts slightly into the layers. Serve warm, spooning up some of that caramel sauce from the bottom of the pan for extra decadence.