Gather ingredients and equipment. Melt the butter and set aside. Warm the milk in the microwave for 40 seconds. Check temperature with a thermometer to ensure 110℉-115℉.
With a whisk, combine warmed ¾ cup milk, 2¼ tsp yeast and ¼ cup sugar in the bowl of a stand mixer and let sit for about 5 minutes. Mix in the egg and egg yolk and melted and cooled ¼ cup butter.
With a wooden spoon, mix in 370 grams of bread flour and ¾ tsp of salt until combined. Using the dough hook of the stand mixer, have the machine knead the dough on medium-low for approximately 8 minutes. The dough should form a ball.
Transfer dough to a large oiled, glass bowl and cover with a towel for 1 - 2 hours. The dough should double in size.
Prepare the 9" x 9" baking pan by spraying baking spray or lining it with parchment paper.
Transfer dough to a well-floured surface and roll out to a rectangle approximately 14" x 9". Spread the ¼ cup of softened butter onto the dough. Cover with brown sugar and cinnamon. If using unsalted butter, sprinkle a little salt over the filling as well.
Starting from the 9" side, roll the dough as tight as possible. It should be easy to work with. Make 9 rolls by slicing 1" pieces using a bread knife. Place the rolls into the 9" x 9" baking pan.
Same day baking: Cover with plastic wrap or a towel and allow the rolls to rise again for about an hour before baking.
Next day baking: Cover with plastic wrap and place in the fridge overnight. In the morning, take the rolls out of the fridge as soon as you wake up and allow them to come to room temperature before baking. From the refrigerator, we keep ours on the counter for 2 to 4 hours before baking. This allows them to fully rise before getting in the oven.
Preheat the oven to 350℉. Remove plastic wrap and bake for 20-25 minutes until they are golden brown on top. While rolls are baking, make the frosting by mixing the ingredients with an electric mixer. Allow the rolls to cool for 5-10 before covering with the frosting.
Enjoy! If they last more than a day, store in a tight container in the fridge for up to a week.