Warm the milk slightly, then stir in the sugar and yeast. Let it sit for 10 minutes until the yeast becomes frothy.
In a large bowl, combine the flour and salt.
Add the yeast mixture, egg, and melted butter, then knead into a soft, elastic dough.
Cover the dough with a kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
Turn the dough out onto a lightly floured surface and divide it into 8 equal pieces. Shape them into dumplings.
In a large skillet with a lid, melt the butter, then pour in the milk.
Place the dumplings in the skillet, making sure they don't touch, then cover with the lid.
Cook over medium heat for about 15 minutes without lifting the lid.
When the milk has evaporated and the bottoms of the dumplings are golden brown, turn off the heat and let them rest for a few minutes.
Serve warm with vanilla sauce or jam.
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