Heat the milk, butter, sugar, and salt in a small saucepan over low heat until the butter melts. Remove from the heat and whisk in the eggs.
Combine the flour and yeast in the bowl of a stand mixer fitted with a dough hook. Add the milk mixture and knead on low speed until the dough is smooth and elastic, about 10 minutes.
Cover the bowl and let the dough rise at room temperature until doubled in size, 1 to 1½ hours.
Spoon the chocolate spread into 8 small portions on a parchment-lined tray and freeze until firm.
Divide the dough into 8 equal pieces. Flatten each piece, place a frozen chocolate portion in the center, and pinch the dough closed. Roll into smooth balls.
Arrange the buns in a greased muffin pan. Cover and let rise until puffed, about 45 minutes.
Meanwhile, whisk the melted butter, brown sugar, egg, flour, and dissolved coffee until smooth. Transfer the mixture to a piping bag.
Pipe the coffee topping over the buns.
Heat the oven to 350°F/180°C. Bake until golden brown and cooked through, about 30 minutes.
Let cool slightly before serving.