Start by combining warm milk, sugar, melted butter, salt, yeast, and red food coloring in a mixing bowl. Add eggs and sour cream, then gradually mix in flour and cocoa powder. Knead until the dough becomes smooth and elastic.
Transfer the dough to a lightly greased bowl and cover it. Let it rise in a warm place until it doubles in size, which usually takes one to two hours.
Once risen, roll the dough out into a large rectangle on a floured surface. Spread softened butter evenly, then sprinkle the brown sugar and cinnamon mixture over the top.
Roll the dough tightly into a log and cut it into 12 equal slices. Place them in a greased baking dish, cover, and allow them to rise again for about one hour.
Preheat the oven to 375°F (190°C). Bake the rolls for 35–40 minutes until they are puffed and lightly golden on top.
While the rolls cool slightly, beat butter and cream cheese until smooth. Add powdered sugar, vanilla, and milk, adjusting the consistency as needed. Spread over warm rolls.