Warm the milk and 2 tablespoons butter until the butter is just melted and the mixture is lukewarm. Combine the pumpkin, sugar, ¼ teaspoon nutmeg, and salt in a large bowl. Whisk in the milk mixture, egg, and yeast.
Mix in 1 cup of flour, then add 1⅔ cups more, kneading to form a soft, slightly tacky dough. Perform the windowpane test to ensure proper kneading.
Let the dough rise in a greased bowl, covered, in a warm place until doubled in size, about 2 hours.
Grease a baking dish. Punch down the risen dough and roll out into a 10x14-inch rectangle on a floured surface.
Spread 6 tablespoons softened butter over the dough. Mix the brown sugar and spices, then sprinkle over the butter. Roll up the dough and cut crosswise into rolls. Place in the prepared baking dish.
Cover the rolls and let rise until doubled, about 1 hour. Preheat the oven to 350°F.
Bake the rolls for 22-28 minutes, covering with foil if browning too quickly. Let cool slightly before icing.