In a small bowl, bloom the yeast in lukewarm milk with a pinch of sugar. Let it stand for 10 minutes, until foamy.
In a large bowl, whisk together the flour, remaining sugar, salt, and grated lemon zest.
Add the bloomed yeast, eggs, and vanilla extract to the flour mixture and mix until combined.
Knead the dough for about 10 minutes, then gradually add the softened butter and work it in completely.
Cover the dough and let it rise in a warm place for 1.5 hours, or until doubled in size.
On a lightly floured surface, roll out the dough to about 2 cm thickness. Use a doughnut cutter or a glass to cut out circles.
Place the cut dough pieces on a baking sheet lined with parchment paper and let them rise for another 30 minutes.
In a large pot, heat the oil to about 340°F (170°C). Test the oil by dropping in one doughnut: if it immediately floats to the surface and gently sizzles, the oil is ready.
Fry the doughnuts for 2-3 minutes per side, until golden brown. Use a slotted spoon to remove them and place them on paper towels to absorb excess oil.
Sprinkle the cooled doughnuts with powdered sugar and serve fresh.
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