Start by warming the milk to about 110°F, which is usually achieved by microwaving it for 40-45 seconds. It should feel like warm bath water.
In the bowl of an electric mixer, sprinkle yeast over the warm milk. Add sugar, egg, egg yolk, and melted butter. Mix until well combined, then stir in flour and salt until a dough starts to form.
Knead the dough with a dough hook on medium speed for 8 minutes until it forms a ball and is slightly sticky.
Place the dough in a well-oiled bowl, cover with plastic wrap and a warm towel, and let it rise until doubled in size, about 1 to 1 1/2 hours.
Roll out the dough on a floured surface into a 14x9 inch rectangle. Spread softened butter over the dough, leaving a small margin on one edge.
Mix brown sugar and cinnamon together and sprinkle over the buttered dough. Press the mixture into the butter.
Roll the dough tightly from the short edge and seal the seam. Trim the ends and cut the roll into 9 pieces.
Place the rolls in a greased and lined 9x9 inch baking pan, cover, and let rise for another 30-45 minutes.
Preheat the oven to 350°F. Bake the cinnamon rolls for 20-25 minutes until slightly golden brown. Cool for 5-10 minutes before frosting.
For the frosting, beat together cream cheese, butter, powdered sugar, and vanilla extract until smooth. Spread over the warm rolls and serve.