Dough. Lightly grease a bowl with non-stick spray and set aside. Place cut up potato in a small pot of water and bring to a boil. (Do not salt the water.) Cook for about 10 minutes until a fork inserted into the potato pieces can be inserted easily. Drain the potatoes and mash with a fork. Measure out 146 g. (2/3 cup) and allow to cool slightly.
While the potatoes are cooking, in a small pot or saucepan, melt the butter over low heat and allow to cool.
Heat milk to slightly above body temperature (generally between 105°F and 110°F) in a small pot or in the microwave at 20 second intervals, stirring after each interval. Transfer warmed milk to the bowl of your mixer. Sprinkle the yeast over the milk. Add a pinch of the sugar. Using a fork, stir the mixture to distribute the sugar and to ensure all the yeast granules are hydrated. Allow to sit for approximately 3-5 minutes until the mixture gets foamy.
Add the melted butter, mashed potato, sugar, egg, and bread flour. With the dough hook attachment, knead on a medium speed for about 7 minutes, stopping the mixer every couple of minutes to scrape down the sides of the bowl. With the mixer on a lower speed, slowly add the salt. Increase mixer speed to medium and continue kneading for an addition 7-8 minutes, again stopping the mixer every couple of minutes to scrape down the side of the bowl. If the dough seems too wet and sticky, add 1 tbsp. of flour. However, the dough should not entirely clean the sides of the bowl as it is kneaded and should be slightly sticky.
Place dough into the lightly greased bowl, turn it over once. Cover with plastic wrap and allow to double in size, approximately 1½ hours. (Alternatively, the dough can be refrigerated, covered with plastic wrap, for up to 2 days.).
Prepare a 9 inch by 13 inch pan by either lining it with parchment paper, spraying it with non-stick spray, or buttering it.
Filling. In a bowl, stir the brown sugar with the cinnamon and salt. Set aside.
Punch down the dough, giving it two or three kneads. Roll it out to a rectangle, at least 19 inches long and about 14 inches wide. Have the dough facing you so that the width is 19 inches and the 14 inches runs vertically.
Spread the room temperature butter all over the dough, leaving ¼ inch border across the top unbuttered.
Sprinkle the filling all over the butter and spread it evenly.
Roll the dough up, away from you, in a tight roll. If the dough is sticking, flour a bench scraper to slide under the dough as you roll, or, if you are rolling on parchment paper, use the parchment to lift the dough up and roll it onto itself. Pinch the seam closed. Place log, seam side down, on a piece of parchment paper and cover loosely with plastic wrap. Place in refrigerator for 30 to 60 minutes, just until the dough is somewhat firm.
Trim about ½ inch from one end. Using a ruler, cut 12 rolls, each 1½ inches wide. Place them in the prepared pan, 4 across the 13 inch side, and 3 along the 9½ inch side. There should be space between the rolls. Cover the pan with plastic wrap completely so no air can get in. Refrigerate overnight. (You should have scraps left over from the two ends. Grease a small ramekin, knead the scraps together, and place the ball of scraps in the prepared ramekin. Cover with plastic wrap and refrigerate overnight with main batch of rolls.)
The next morning, remove the pan (and the ramekin) from the refrigerator. Rest them on the counter at room temperature for approximately 1½ hours. They should almost double in size.
Approximately 30 minutes before you plan to bake, preheat the oven to 350°F.
Bake the rolls for 35-40 minutes. The tops should be golden brown, and they should not jiggle when you move the pan. If you are using a thermometer, the inside temperature should be about 210°F. (The lone roll in the ramekin will likely be done approximately 5 minutes before the rest of the rolls.) To take the temperature of the rolls, I highly recommend this thermometer from ThermoWorks. (I literally use it to take the temperature of anything I make, from ice cream to breads to tempering chocolate.)
Glaze. After the rolls have been in the oven for 30 minutes, whisk together all the ingredients for the glaze, starting with just 2 tbsp. of the milk. If the glaze is too thick, add a little more milk until you reach a smooth, spreadable consistency. When the rolls come out of the oven, dollop a spoonful of the glaze on each roll. Spread the glaze all over. Allow to cool at least 20 minutes before serving.