Warm the milk slightly, then stir in the sugar and yeast. Let it stand for 10 minutes, until frothy.
In a large bowl, sift the flour, then add the salt, vanilla extract, and egg.
Pour in the yeast mixture and the melted butter, then 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 lightly floured surface, roll out the dough to about 1.5 cm thickness, then use a glass or donut cutter to cut out circles. Cut out the centers with a smaller circular cutter to create ring donuts.
Place the donuts on baking paper, cover them, and let them rise for another 30 minutes.
Heat the oil in a deep pan to about 340°F (170°C).
Fry the donuts for 2-3 minutes per side, until golden brown. Remove them with a slotted spoon and place them on paper towels to absorb the excess oil.
While still warm, toss them in granulated sugar, then serve.
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