Warm the milk until lukewarm, then mix in the yeast and a little sugar. Let it stand for 10 minutes, until foamy.
In a large bowl, combine the flour, remaining sugar, vanilla sugar, salt, and cinnamon.
Add the proofed yeast mixture, beaten egg, and melted butter. Knead the dough until it is smooth and elastic.
Cover the dough with a kitchen towel and let it rise in a warm place for 1 hour, or until doubled in size.
Roll out the risen dough to about 1 cm thick. Use a glass or doughnut cutter to cut out circles.
Preheat the grill to medium heat (about 350°F or 180°C) and place parchment paper or a BBQ baking sheet on the grill grates.
Place the doughnuts on the grill and cook for 3-4 minutes on each side, or until golden brown.
Immediately after removing from the grill, brush the tops with a little honey for a delicious glaze.
Let cool for a few minutes, then serve fresh, sprinkled with powdered sugar or with fruit.
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