Warm milk to around 110°F. Transfer warm milk to the bowl of an electric mixer and sprinkle yeast on top. Add sugar, egg, egg yolk, and melted butter. Mix until well combined. Stir in flour and salt with a wooden spoon until a dough begins to form.
Place dough hook on stand mixer and knead dough on medium speed for 8 minutes. Dough should form into a nice ball and be slightly sticky. If it's too sticky, add 2 tablespoons more bread flour. Alternatively, knead by hand for 8–10 minutes on a well-floured surface.
Transfer dough ball to a well-oiled bowl, cover with plastic wrap and a warm towel. Allow dough to rise for 1 to 1½ hours, or until doubled in size.
Transfer dough to a well-floured surface and roll out into a 14×9 inch rectangle. Spread softened butter over dough, leaving a ¼ inch margin at the far side.
In a small bowl, mix together brown sugar and cinnamon. Sprinkle mixture over the buttered dough and rub the brown sugar mixture into the butter.
Tightly roll dough up starting from the 9-inch side. Place seam side down and seal the edges. Cut off about an inch from each end, then cut into 1-inch sections with a serrated knife or floss. You should get 9 large pieces.
Place cinnamon rolls in a greased 9×9 inch baking pan or round 9-inch cake pan lined with parchment paper. Cover with plastic wrap and a warm towel and let rise again for 30–45 minutes.
Preheat oven to 350°F. Remove plastic wrap and towel and bake for 20–25 minutes, until just slightly golden brown on the edges. Allow to cool for 5–10 minutes before frosting.
For the frosting, combine cream cheese, butter, powdered sugar, and vanilla extract in an electric mixer. Beat until smooth and fluffy. Spread over warm cinnamon rolls and serve immediately.