To make the dough, dissolve the yeast and 1 tablespoon of the sugar in 1/2 cup lukewarm water. Set aside for about 5 minutes until foamy. Meanwhile, warm the milk in a saucepan. Add the butter, remaining sugar, and salt, and stir until dissolved. Remove from the heat.

Sift 3 cups of flour into a large bowl. Add the yeast, warm milk mixture, and the egg, and stir to make a dough. Knead for 10 minutes on a lightly floured board, incorporating more dough as necessary, until the dough is soft, silky, and pliable. Form into a ball. Place in a buttered bowl and turn to coat the dough all over. Cover with plastic wrap and leave to rise until twice its original size. This will take about 2 hours. At this point, the dough can be left overnight in the fridge, ready to use the next morning. When well risen, punch down the dough, then leave to rest for 10 minutes. Meanwhile, butter a baking sheet. Prepare filling by mixing the sugar, cinnamon, and nuts in a small bowl.

Roll out the dough into a rectangle measuring about 12 x 9 inches. Brush with the melted butter. Sprinkle the filling mixture over the dough, pressing it in slightly. Roll up the dough like a jelly roll, starting at one long side. Using a sharp knife, cut the roll across into 12 equal slices. Arrange the rolls, cut-side up, on the prepared baking sheet. Allow to rise, covered, for at least 40 minutes until doubled in size.

Preheat the oven to 400 degrees F. Bake for 15-20 minutes, or until well risen and golden. While the buns are in the oven, prepare the glaze (and get the coffee started). Sift the confectioners' sugar into a bowl. Whisk in the melted butter and enough milk to make a thick but pourable mixture. When the buns are ready, remove from the oven. Pour the glaze over them and leave to set for a few minutes before devouring.