Warm the milk to 110°F. Combine warmed milk, melted butter, sugar, and yeast in a large bowl, allowing it to stand for 10 minutes until frothy.

Whisk the room-temperature egg in a separate bowl and set aside. Incorporate the egg into the yeast mixture and mix until well combined.

Gradually add flour and salt, stirring until a dough begins to form.

Knead the dough for about 7 to 8 minutes with a stand mixer or 8 to 10 minutes by hand until it feels slightly sticky.

Shape the dough into a ball, place in a lightly greased bowl, cover with plastic wrap and a damp towel, and let it rise for about 1 hour until doubled in size.

Meanwhile, mix brown sugar and cinnamon in a small bowl to prepare the filling.

After the dough has risen, roll it out into a 14 x 9 inch rectangle on a well-floured surface.

Spread softened butter over the rectangle, leaving a ¼ inch border. Sprinkle the cinnamon-sugar mixture generously over the butter.

Roll the dough tightly from one short edge and seal the edge by pinching it closed. Slice the roll into 9 even pieces.

Arrange the cinnamon rolls in a lined 13 x 9 inch baking dish, ensuring they are slightly touching. Cover and let rise for about 40 minutes.

Preheat oven to 350°F. Remove coverings and bake for 20 to 25 minutes, checking for golden edges from the 18-minute mark.

Let the rolls cool for 10 minutes on a rack.

Mix frosting ingredients until smooth. Spread over the warm rolls and serve immediately.