Dough. [If using instant yeast, skip to Step 3 and add the yeast and the milk with the rest of the ingredients.] In microwave at 20 second intervals, stirring after each interval, warm milk to body temperature, generally between 105°F and 110°F. Alternatively, you can heat the milk in a saucepan on a low flame, stirring constantly. Either way, do not let the milk get above 115°F. (If you do, just let it sit out at room temperature for a couple of minutes constantly checking it until it cools to body temperature.)
Place warmed milk into bowl of electric mixer. Sprinkle yeast and approximately 1 tsp. of the sugar over the milk. Using a fork, stir the mixture to distribute the sugar and hydrate the yeast granules. Allow to sit for approximately 3-5 minutes until the mixture gets foamy.
Add the room temperature egg, the room temperature egg yolk, and the vanilla to the mixture. Using the dough hook, turn the mixer on at a low speed and add the rest of the sugar. With the mixer running at the lowest speed, add the flour. As the flour starts to get absorbed by the liquid mixture, raise the mixer to a medium speed. You may need to turn the mixer off once or twice to wipe down the sides of the bowl with a rubber spatula. Allow to knead for approximately 3 minutes. The dough should clean the sides of the bowl and be tacky, even a bit sticky. If it is too wet (sticking to the sides and bottom of the bowl), add some flour, no more than 1 tbsp. at a time. If the mixture is too dry, add a little milk, 1 tsp. at a time.
Lower the mixer speed to low (1 or 2 on the KitchenAid) and slowly add the salt, allowing it to knead into the dough, an additional 3 minutes. Stop the mixer and take a small piece (approximately 1 tsp.) of dough and stretch it between your fingers. It should form a “window pane”, meaning it should get thin enough that it is translucent without it ripping. If it is not at that point yet, continue kneading in the machine at 1 minute intervals. (It may just need an additional minute or two to reach the window pane stage. If it still isn’t there yet, check to see if the dough is too dry. If it is, add ½ tsp. milk and knead another minute.)
With the mixer running at medium speed, add the butter. It is best to add it one piece at a time, waiting until it is fully integrated into the dough before adding the next piece. You may need to turn off the mixer from time to time to push the slab of butter back into the range of the dough and the hook. This process will take approximately 5-10 minutes. Once all the butter is incorporated, the dough will look shiny and be slightly sticky. (If, after approximately ten minutes, the butter is still not fully incorporated, do not panic. Just move to the next step – having some unincorporated butter is not problematic in this recipe. That said, the key to incorporating the butter is nothing more than patience.)
Lightly grease a bowl with a little butter or nonstick cooking spray. Form dough into a ball and place it in the bowl. Turn the dough over to coat it in the butter or nonstick cooking spray and cover tightly with plastic wrap. Let the dough sit in a warm area on the counter until doubled in size, approximately one hour.
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 metal spatula 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. Close the seam at the top.
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 plenty of space between the rolls. Cover the pan with plastic wrap completely so no air can get in. Refrigerate overnight.
The next morning, remove the pan from the refrigerator. Rest them on the counter for approximately 1½ hours. They should almost double in size.
About an hour into the last rise, 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.
Icing. After the rolls have been in the oven for 30 minutes, whisk together all the ingredients for the icing, starting with just 2 tbsp. of the milk. If the icing 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 icing on each roll. Spread the icing all over. Allow to cool at least 20 minutes before serving.