In a mixing bowl, combine warm milk, sugar, and yeast. Let it sit for 5 minutes until frothy.

Add melted butter, egg, and salt to the yeast mixture. Gradually mix in flour until a dough forms.

Knead the dough on a floured surface for about 5 minutes until smooth. Place in a greased bowl, cover, and let rise for 1 hour.

In a small bowl, mix together brown sugar, ground coffee, and cinnamon.

Roll out the risen dough on a floured surface into a rectangle. Spread softened butter over the dough, then sprinkle the filling evenly.

Roll the dough tightly into a log and slice into 12 equal pieces. Place rolls on a greased baking sheet.

Cover and let rise for another 30 minutes. Preheat the oven to 350°F (175°C).

Bake for 20-25 minutes until golden brown. Remove from oven and let cool slightly.

In a bowl, whisk together powdered sugar, milk, and vanilla until smooth. Drizzle over warm rolls.