Place potato in a small pot of water and bring to a boil. Cook until fork-tender, 10 to 12 minutes; drain. Pass through a ricer or mash with a fork until smooth (to yield ⅔ cup); let cool slightly.

Heat milk until warm to the touch (110°F on a thermometer). Transfer to the bowl of a stand mixer; add yeast. Let stand until foamy, about 5 minutes.

Stir in butter, granulated sugar, potato, egg, flour, and salt until a ragged dough forms. Place bowl on mixer fitted with the dough-hook attachment; mix on medium-low speed, scraping sides and removing dough from hook a few times, until soft and pliable, about 15 minutes.

Transfer to a lightly floured surface; knead into a ball. Spray mixing bowl with Pam. Transfer dough into bowl (dough will be very soft) and cover with plastic wrap. (Dough can be covered and refrigerated at this point up to 2 days). Let rise in a warm place until almost doubled, 1 hour and 10 minutes to 1 ½ hours.

Brush a 9x13-inch glass baking dish with butter; set aside.

In a medium bowl, stir together brown sugar, cinnamon, and salt.

Punch down dough; transfer to a lightly floured surface. Roll out to a 14-by-18-inch rectangle.

Spread evenly with butter, leaving a ½-inch border on far long edge; top with brown-sugar mixture. Starting at long edge closest to you, roll dough away from you into a tight jelly-roll shape. Transfer, seam-side down, to a parchment-lined baking sheet and refrigerate until firm, about 30 minutes.

Trim ends, then cut roll crosswise into twelve 1 ½-inch-thick rounds with dental floss. Arrange, cut-sides up, in prepared dish; cover with plastic wrap. Refrigerate at least 8 hours, or overnight.

Preheat oven to 350°F. Remove rolls from refrigerator and place in a warm spot until almost doubled in size, 1 ½ - 2 hours.

Remove plastic and bake until browned on top and a thermometer inserted registers about 210°F, about 30 - 35 minutes. Let cool in dish on a wire rack, 10 minutes.

Meanwhile, whisk together confectioners' sugar, milk, vanilla and kosher salt. Spread rolls with glaze; let cool 15 to 20 minutes more before serving.