In a small bowl, combine the warm water and yeast. Let stand for 5–10 minutes, until foamy.
In the bowl of a stand mixer fitted with the paddle attachment, combine the yeast mixture, milk, melted butter, brown sugar, granulated sugar, salt, and egg. Gradually mix in the flour, 1 cup at a time, until a soft dough forms.
Transfer the dough to a lightly greased bowl, turning once to coat. Cover with plastic wrap or a clean towel and let rise for 1 to 1½ hours, or until doubled in size.
Punch down the dough and knead for 1–2 minutes on a lightly floured surface. Roll into a rectangle about 24 x 15 inches.
Stir together the melted butter, brown sugar, and cinnamon for the filling. Spread evenly over the dough.
Starting with the short side, roll the dough tightly into a log. Slice into 15–18 rolls using unflavored dental floss or a sharp knife.
Arrange the rolls snugly in a greased 9x13-inch baking dish. Cover and let rise for 30–60 minutes, or until nearly doubled.
Bake at 375°F for 20–25 minutes, or until the centers are cooked through and the tops are golden brown.
While the rolls bake, beat together the cream cheese, softened butter, powdered sugar, vanilla, and milk until smooth.
Spread the frosting over the warm cinnamon rolls and serve immediately.
Notes
Baking time may vary depending on the size of your cinnamon rolls.
If the tops brown before the centers are fully baked, loosely tent the pan with aluminum foil for the remaining bake time.
For the cleanest slices, use unflavored dental floss instead of a knife when cutting the dough.