Warm the milk, then dissolve the sugar and yeast in it. Let it stand for 10 minutes, until foamy.
In a large bowl, sift the flour, then add the salt, vanilla extract, eggs, melted butter, and rum or vodka.
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, until doubled in size.
On a floured surface, roll out the dough to about 2 cm thick, then use a glass or donut 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 saucepan 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 pączki for 2-3 minutes on each side, until golden brown. Remove them with a slotted spoon, and place them on paper towels to absorb excess oil.
Using a piping bag, fill the donuts with jam, then sprinkle with powdered sugar.
Serve fresh, or store in an airtight container to keep them soft.
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