In a small pot, heat the milk over low heat until warm but not boiling. Remove from heat and stir in the 2 tablespoons of butter until melted. Allow to cool slightly. In a large bowl, combine the warm milk mixture with pumpkin puree, granulated sugar, salt, beaten egg, and yeast. Stir until well combined.
Gradually add the all-purpose flour and bread flour to the wet ingredients, mixing until a soft dough forms. The dough should be slightly sticky but manageable.
Turn the dough onto a lightly floured surface and knead for about 5-7 minutes until smooth and elastic. Place into a greased bowl, cover with a clean towel, and let rise in a warm place until doubled in size, about 1 hour.
Prepare the filling by mixing together the brown sugar, ground cinnamon, and 2 tablespoons of melted butter in a bowl.
Once the dough has risen, punch it down and roll it out on a floured surface into a rectangle about 1/4 inch thick. Spread the cinnamon filling evenly over the dough.
Starting from one long edge, carefully roll the dough tightly into a log. Slice the log into 12 equal pieces. Place in a greased baking dish, cut side up.
Cover with a towel and let the rolls rise again in a warm place until puffed and nearly doubled, about 30 minutes.
Preheat your oven to 350°F (175°C). Bake for about 20-25 minutes, or until golden brown. Remove and let cool slightly.
In a medium bowl, combine the melted butter, brown sugar, 2 tablespoons of milk, vanilla extract, and salt. Mix until smooth. Gradually whisk in the powdered sugar until desired consistency is reached.
Drizzle the caramel icing generously over the warm rolls. Let sit for a few minutes to allow icing to set slightly before serving.