In the bowl of a stand mixer fitted with a dough hook, combine the flour, sugar, pie spice, yeast, and salt on low speed until well mixed.

In a large measuring jug with a pour spout, beat together the milk, pumpkin puree, butter, and egg, then pour this into the dry ingredients and mix on low speed until the mixture begins to pull away from the sides of the bowl. If the dough seems a bit wet, add the remaining flour, one tablespoon at a time, until the dough just starts to pull away but is still quite soft and a bit sticky.

Oil a large bowl and your hands, and then transfer the dough to the bowl. Cover and let rest in a warm place until doubled in size, about 90 minutes.

Once the dough has risen, turn it out onto a floured surface and roll the dough into a roughly 12x18-inch (30x46cm) rectangle with the longer side closest to you. It should be a long roll, not a squat roll.

Spread the pumpkin butter evenly over the surface of the dough, then, starting with a long edge, roll the dough into a tight log.

With a serrated knife, and using a sawing motion, cut the log in half. Cut each half into 2 equal pieces so that you have 4 equally sized small logs. Cut each small log into 3 equally sized pieces. You should have 12 equal-sized pieces.

Butter a 9x13-inch (23x33cm) baking dish and place the rolls cut side up in the dish.

Cover with plastic wrap of a lightly floured tea towel and let rise until doubled, about 45 minutes.

In a medium bowl, whisk together the powdered sugar and the cream cheese until smooth. Stir in the butter until combined, then add in the maple syrup and vanilla extract. Set aside.