Combine the warm milk, melted butter, and yeast in a large mixing bowl. Let sit for 5 minutes until the yeast is frothy and activated, indicated by bubbly foam on the surface.
Add the egg, granulated sugar, and a pinch of salt to the yeast mixture. Whisk until well combined, then gradually add the flour, mixing with a spatula or dough hook until a soft, slightly sticky dough forms.
Knead the dough on a floured surface for about 5 minutes until smooth and elastic. Place the dough in a greased bowl, cover with a damp cloth, and let rise in a warm area for 1 hour or until doubled in size.
Roll out the risen dough into a rectangle approximately 12x8 inches using a floured rolling pin. Spread the softened butter evenly over the surface, then sprinkle with brown sugar mixed with cinnamon.
Carefully roll the dough tightly starting from the longer edge, then slice into 8 equal pieces. Arrange the rolls in a greased baking dish, leaving space between each, and let rise for an additional 20 minutes until puffy.
Bake the rolls in a preheated oven at 350°F for 22-25 minutes until golden brown and cooked through. Prepare the icing by mixing powdered sugar with milk until smooth, then drizzle over hot rolls before serving.
