In a large bowl, combine warm milk, yeast, sugar, pumpkin purée, melted butter, and egg. Mix until smooth.
Stir in the flour, pumpkin pie spice, and salt until a shaggy dough forms.
Knead on a lightly floured surface for 8–10 minutes (or use a stand mixer with dough hook) until smooth and elastic.
Place dough in a greased bowl, cover, and let rise in a warm place for 1–1.5 hours, until doubled in size.
In a bowl, mix together brown sugar, cinnamon, and pumpkin pie spice.
Roll dough into a 12×16-inch rectangle.
Spread softened butter evenly over the dough.
Sprinkle the sugar-spice mixture evenly over the top.
Roll tightly from the long edge to form a log, then slice into 12 rolls.
Arrange in a greased 9×13 baking dish, cover, and let rise 30 minutes.
Preheat oven to 350°F and bake 20–25 minutes, until golden.
Whisk icing ingredients together, drizzle over warm rolls, and serve.