Combine the milk, butter, sugar, and salt. Stir until sugar is dissolved.
Mix in egg and egg yolk.
Add 1 cup of the flour mixed with the yeast. Add more flour until the dough is soft and pulls away from the sides of the mixer bowl. It can take slightly more or less flour.
Finish by adding the remaining three tablespoons soft butter, one tablespoon at a time until completely incorporated.
Form dough into a ball and place in a greased bowl. Cover with plastic wrap. Place in a warm area for 30 minutes.
While raising, mix together the butter, brown sugar, and cinnamon.
Grease a 12-cup muffin tin and set aside.
After the dough has rested 30 minutes, roll out on a floured surface to a roughly 12x18 rectangle.
Spread the filling mixture evenly, leaving a ¼-inch border. Tightly roll up and pinch seam. Cut into 12 equal portions and place in a prepared muffin pan. Rest rolls in a warm area for 30 minutes.
Preheat oven to 350°F.
After resting 30 minutes, bake the rolls in preheated oven 22-28 minutes. When done, the rolls will be puffed and golden brown.
Cool in pan for 10 minutes while preparing the frosting.
Over butter over medium heat in a 4-quart saucepan and continue cooking until browned and nutty, 3-5 minutes. Remove from heat and add powdered sugar, vanilla extract, and whole milk. Whisk until smooth, adding more milk as needed.
Spoon over warm rolls. The frosting will create a crust as it cools.