Heat the milk so that it is lukewarm but not hot. Pour ½ cup of milk into a cup, add a teaspoon of sugar, add active dry yeast, and stir. Leave it in a warm place until small bubbles form on the surface of the activated yeast. This takes about 10-15 minutes.
Meanwhile, melt the butter and let it cool down; it can't be hot.
In a large mixing bowl, place the flour, sugar, the rest of the lukewarm milk, melted butter, a pinch of salt, and an egg. Add milk with activated yeast.
Make a smooth dough: Start stirring the ingredients with a fork or wooden spoon; when combined, dump the mass on the work surface. Knead the dough properly. It requires a bit of physical effort to make the dough smooth and slightly sticky. It takes about 10 minutes. If the dough is too sticky during processing, add a little flour.
Let the dough rise in a warm place until doubled in size. It takes about one hour.
Divide the dough into approximately equal-sized pieces. If you have a kitchen scale, weigh each piece of dough per bun to 3 ounces (85 grams). Roll each piece with a rolling pin into a patty.
The other method (without a scale) is as follows: Roll out the dough on the worktop roughly into a rectangular shape. The dough should be about as thick as a straw. Using a pastry cutter or knife, divide the rolled-out dough into squares or rectangles approximately 4x4 inches (10x10 cm) in size; place about one tablespoon of filling in the middle of each square.
Fold the edges of the dough over the filling and seal well. Finally, gently roll the finished bun between your palms to get a regular shape.
Grease a baking dish about 13x9 inches (33x23 cm) with butter. Melt 1/3 stick of butter (37 g) in a saucepan and set aside to cool. Brush the baking tray with it.
Place the prepared buns one at a time, seam-side down, in the baking dish, brushing each side with melted but not hot butter. Keep going until you've used up all the dough.
Finally, brush the buchty tops with the remaining butter. Let them sit in a warm place for about 20 minutes; they will rise a little more.
Bake in a preheated oven at 325°F (165°C) for about 40 minutes, until golden brown.
Melt two tablespoons of butter (25 g), add two tablespoons of rum, and mix. Once the buns are fresh out of the oven, brush their surface generously with the butter-rum mixture.