In a large bowl, mix flour, sugar, yeast, and salt. Add warm milk, melted butter, and the egg. Stir until a soft dough forms, then knead for 5–7 minutes until smooth and elastic. Cover and let it rise for 1 hour or until doubled in size.
Prepare the filling.
In a small bowl, combine brown sugar and cinnamon. Roll out the dough into a rectangle, about ¼ inch thick. Spread softened butter evenly, then sprinkle the sugar mixture on top.
Shape and cut.
Roll the dough tightly into a log and slice into 9–12 even rolls. Place them in a greased baking dish, cover, and let them rise for another 30 minutes.
Bake.
Preheat the oven to 350°F (175°C). Bake for 20–25 minutes, until the rolls are golden and soft in the center.
Make the glaze.
Whisk together powdered sugar, cream cheese, milk, and vanilla until smooth. Spread over warm rolls and let it melt slightly into the swirls.