In a small bowl, combine the yeast with a little lukewarm milk and a tablespoon of sugar. Let it sit for 5-10 minutes, until foamy.
In a large bowl, sift the flour, add the sugar, salt, grated lemon zest, and vanilla extract.
Make a well in the center of the flour, pour in the yeast mixture, the eggs, and the remaining lukewarm milk, then mix to combine.
Add the melted butter and knead the dough for 10-15 minutes, until smooth and elastic.
Cover the dough and let it rise in a warm place for 1 hour, or until doubled in size.
On a floured surface, roll out the dough to about 1 cm thickness, then cut into small squares or diamonds.
Place the dough pieces on a floured baking sheet, cover them, and let them rise for another 30 minutes.
In a saucepan, heat the oil to 340-355°F (170-180°C). The oil is ready when a small piece of dough slowly sizzles in it.
Fry the dough pieces until golden brown, about 2-3 minutes, turning them occasionally.
Remove the fried doughnuts and place them on paper towels to absorb excess oil.
Serve warm, dusted with powdered sugar. Best enjoyed fresh!
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