How to Make It
Prepare the dough: In a small pot, bring the milk just to a boil over medium heat. (Watch closely to ensure that it doesn’t boil over.) Remove from the heat and add the butter, stirring to melt. Transfer mixture to a small bowl and let cool until warm to the touch but not too hot, about 105°F to 110°F. Add the egg and egg yolk 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 your mixer to the dough hook and knead the dough on low speed until smooth and elastic, about 6 minutes. Or, tip the dough onto a work surface and knead by hand until smooth and elastic, about 12 minutes. If you plan to bake the buns later in the day, transfer the dough to a buttered bowl, cover it, and leave it in a warm, draft-free spot until it has doubled in size. If you plan to bake the buns the following day or later in the week, form the dough into a ball, wrap it in plastic wrap, and transfer to the refrigerator for 12 hours or up to 3 days.
Prepare the filling: Combine the brown sugar, cinnamon, and salt in a small bowl. Cover with plastic wrap and set aside.
Prepare the buns: Butter a 12-inch cast iron pan. Place the dough on a lightly floured work surface and roll it into a 12-by-11-inch rectangle. Spread 3 tablespoons of butter evenly over the surface and sprinkle with the reserved filling. Roll the dough into a tight log and pinch the top seam closed. With a serrated knife, slice the log into 10 equal pieces. Arrange them snugly in the pan with the cut sides facing upwards. Cover loosely with plastic wrap, and let rest until the dough has almost doubled again, 1 to 1½ hours. When risen, the dough should look pillowy and the rolls should be touching.
Preheat the oven to 350°F. Uncover the rolls and carefully brush with beaten egg. Transfer pan to the oven and bake until the buns are golden brown and puffed and a thermometer inserted into their center reads 185°F to 190°F, about 20 to 25 minutes. (If you don’t have a thermometer, give the center roll a wiggle—if it’s set in place, the buns are ready to come out. If the middle of the center roll feels soft, give them another few minutes.) Transfer to a rack to cool.
Prepare the frosting. Beat the cream cheese, confectioners sugar, vanilla, and salt until smooth. Add the milk, 1 tablespoon at a time, until you reach the desired consistency.
When buns are cool to the touch, spread each with a smear of frosting and serve.