Preheat oven to 375°F (190°C) and grease a baking pan.
Mix flour, brown sugar, cinnamon, coffee granules, and salt in a bowl.
Combine warm milk, yeast, and a pinch of sugar in another bowl. Let it sit for 5 minutes.
Add melted butter and egg to the yeast mixture. Slowly stir in the dry ingredients.
Knead dough for about 5 minutes until smooth. Let it rise for 10 minutes.
Roll out dough into a rectangle. Spread butter, sprinkle cinnamon-sugar mix, and roll up.
Slice into 12 rolls and place in the greased baking pan.
Bake for 25 minutes or until golden brown.
Whisk together coffee, powdered sugar, and vanilla for glaze. Drizzle over rolls once cooled.