1/2 teaspoon kosher salt
For the filling:
1/4 cup packed dark brown sugar
2 tablespoons ground cinnamon
1/2 teaspoon kosher salt
3 tablespoons unsalted butter, at room temperature
For the frosting:
4 ounces cream cheese, at room temperature
1/4 cup packed dark brown sugar
1/8 teaspoon ground cinnamon
Pinch kosher salt
3 to 5 tablespoons whole milk, at room temperature
To prepare the dough, in a small pot, bring the milk just to a boil over medium heat. Watch closely to ensure that the milk doesn't boil over. Remove from the heat and add the butter to the pot to melt. Add the mixture to a small bowl and let it cool to 105°F to 110°F. (It should be warm to the touch, but not too hot.) Add the egg and sweet potato purée and stir to combine.
In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl with a wooden spoon, combine the flour, sugar, yeast, and salt. Add the warm milk mixture and mix just until combined.
Switch to the dough hook and knead the dough on low speed until smooth and elastic, about 8 minutes. Form the dough into a ball and transfer it to a buttered bowl, cover it, and leave it in a warm, draft-free spot until it has doubled in size, 1 to 2 hours.
To prepare the filling, mix together the brown sugar, cinnamon, and salt in a small bowl. Cover with plastic wrap and set aside.
Butter a 12-inch cast iron pan. Tip the dough out onto a very lightly floured work surface. Roll it into a 12- by 11-inch rectangle. Spread the 3 tablespoons butter evenly over the surface and sprinkle with the reserved filling. Tightly roll up the dough and pinch the top seam closed. With a serrated knife, cut the roll crosswise into 10 equal pieces. Set them in the pan with the topping, spirals facing upward. Cover loosely with plastic wrap, and let them rest until the dough has almost doubled again, about 1 hour to 90 minutes.
Preheat the oven to 350°F. Uncover the rolls and bake until golden-brown and puffed and a thermometer inserted into the center reads 185°F to 190°F, about 20 to 25 minutes.
Meanwhile, prepare the frosting. Beat the cream cheese, brown sugar, vanilla, and salt until smooth. Add the milk, 1 tablespoon at a time, until it is the desired consistency.