Combine the melted butter, brown sugar, granulated sugar, cinnamon, pie spice and salt. The mixture will firm to a spreadable consistency when kept at room temperature until ready to use. Stir right before using.
Line a baking sheet with parchment. Transfer the dough to the baking sheet and pat it out into a 1-inch-thick rectangle. Cover the dough with plastic wrap.
If making the rolls the day of serving, refrigerate the dough for 1 hour or freeze up to 20 minutes. If making the rolls the next day, refrigerate the rolls overnight.
When ready to bake, heat the oven to 375°F and grease a 9x13-inch baking dish with softened butter.
Lightly flour a work surface, and transfer the dough to onto it. Make sure the parchment has been removed from the dough. Lightly flour the top surface of the dough, and roll the dough out into a ½-inch thick rectangle.
Use an offset spatula to spread the filling mixture, to the edge, evenly over the dough. The top surface of the dough should be covered.
Roll the dough, beginning with the long side closest to you, into a tight spiral, keeping the roll tight as you work.
Use a serrated knife or waxed floss to cut the dough log into 1½-inch rolls. Arrange the rolls on the prepared baking dish, leaving ¼-inch between them.
Loosely cover the rolls with plastic wrap and place in a warm place or proofing oven to rise, about 30-45 minutes.
In a small bowl, whisk together 1 egg and 1-tablespoon milk. Brush the top of the rolls with the egg wash and bake at 375°F for 25-30 minutes or until the rolls are golden and the internal temperature registers 190°F.
While the rolls bake, mix together the frosting ingredients and cover with plastic wrap until ready to use.
When the rolls are finished baking, allow them to cool 10 minutes, then drizzle the icing on top.
Serve warm (if possible) or reheat in a 350°F oven (loosely covered with foil) about 15-20 minutes before serving.