Preheat the oven to 350°F. You won’t need it just yet, but it will assist in the rise of your dough. (If you are making these in a warm or humid environment, you can wait to preheat until the 2nd rise).
In a small saucepan, melt the butter and heat the milk to about 110°F. (Check your yeast for a more specific temperature) Pour it into a large bowl with the warm water. Add the yeast and granulated sugar. Stir and let it sit for about 2 minutes.
Stir in the egg, egg yolks, and 2 cups of the flour.
Once the flour is all incorporated, add 2 more cups of flour and the salt. Only add the last 1/2 cup of flour if the dough is too sticky and you can’t work with it.
Knead the dough in the bowl for about 10 minutes. It will be shiny and smooth. Oil the bowl and form a ball with the dough.
Cover the dough with plastic wrap and place it on the preheating stove to rise for 1 1/2 hours.
Punch the dough down and lightly flour a clean work surface. Stretch the dough evenly into a 9 X 13 rectangle with the short side closest to you.
Mix together the brown sugar and cinnamon, and scatter it evenly across the dough, pressing it in a little as you go.
Cut the sheet of dough in half horizontally. Then cut each half into 6 strips. Twist the strips and wind them into a lightly greased donut pan. You’ll lose a fair amount of the filling when you twist. Scoop it up and sprinkle it over the tops. Cover with plastic wrap and place the donuts back on the preheating stove for the second rise which will take about 45 minutes.
Remove the plastic wrap and bake for 20 - 25 mins. While the rolls are baking, whip together the icing ingredients in a stand mixer (or use the whisk attachment on your hand mixer).