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Dough: Brush a large bowl with butter; set aside. 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 2/3 cup); let cool slightly.
Heat milk until warm to the touch (110 degrees 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. Transfer to buttered 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 1/2 hours. Brush a 9-by-13-inch glass baking dish with butter; set aside.
Filling: 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 1/2-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 1/2-inch-thick rounds with a serrated knife. Arrange, cut-sides up, in prepared dish; cover with plastic wrap. Refrigerate at least 8 hours, or overnight.
Preheat oven to 350 degrees. Remove rolls from refrigerator and place in a warm spot until almost doubled in size, about 1 1/2 hours. Remove plastic and bake until browned on top and a thermometer inserted registers about 210 degrees, 30 to 35 minutes. Let cool in dish on a wire rack, 10 minutes.
Glaze: Meanwhile, whisk together cream cheese, confectioners' sugar, milk, vanilla, and salt. Spread rolls with glaze; let cool 15 to 20 minutes more before serving.