Mix flour, sugar, salt, and yeast in a bowl.
Add warm milk, melted butter, and egg.
Knead dough until smooth and elastic.
Let dough rise until doubled in size.
Roll dough into a rectangle on floured surface.
Spread softened butter over rolled dough.
Sprinkle brown sugar and cinnamon evenly.
Roll dough tightly into a log shape.
Cut into 12 equal pieces.
Place rolls on a lined baking sheet.
Bake at 350°F (175°C) for 20 minutes.
Mix powdered sugar, milk, and vanilla for glaze.
Drizzle glaze over cooled rolls.