In a large mixing bowl, combine warm milk, sugar, salt, yeast, egg, and softened butter. Mix until combined, then gradually add flour until a shaggy dough forms.

Transfer the dough to a floured surface and knead for 5-6 minutes, until smooth and elastic.

Divide the dough into 10 equal pieces, roll each into a ball, flatten it, and place a cube of cold butter in the center. Fold to seal.

Place the buns on a baking tray, cover with a towel, and let them rise for 75-90 minutes.

Prepare the coffee topping by beating butter and sugar until fluffy, then adding the coffee solution and flour.

Preheat oven to 350°F (175°C), pipe topping on buns, and bake for 18-20 minutes until golden brown.