Warm the milk, then mix in the sugar and yeast. Let it stand for 10 minutes until foamy.
In a large bowl, sift the flour, then add the salt, vanilla extract, eggs, and melted butter.
Pour in the yeast mixture and knead the dough for 8-10 minutes until smooth and elastic.
Cover the dough with a clean kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 2 cm (¾ inch) thick. Use a glass or doughnut cutter to cut out circles.
Place the circles on baking paper, cover them, and let them rise for another 30 minutes.
Heat the oil in a deep skillet to about 340°F (170°C). If you don't have a thermometer, drop a small piece of dough into the oil – if it slowly rises and turns golden brown, the oil is at the right temperature.
Fry the doughnuts for 2-3 minutes on each side until golden brown. Remove with a slotted spoon and place on paper towels to drain excess oil.
Using a piping bag, fill the doughnuts with jam, then dust with powdered sugar.
Serve fresh or store in an airtight container to keep them soft.
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