In a small bowl, combine the yeast with a little lukewarm milk and a spoonful of sugar. Let it stand for 5-10 minutes, until it foams and becomes frothy.
In a large mixing bowl, sift together the flour, salt, and remaining sugar. Add the grated lemon zest and vanilla extract.
Make a well in the center of the flour mixture, and pour in the yeast mixture, eggs, and lukewarm milk. Start to combine the dough.
Add the melted butter and knead the dough for 10-15 minutes, until it is smooth and elastic.
Cover the dough with a kitchen towel and let it rise in a warm place for 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 1 cm thickness. Cut out circles with a diameter of 6-8 cm.
Place a small amount of jam on half of the circles, then cover with another circle. Press the edges together to seal well.
Place the filled pastries on a floured tray, cover them, and let them rise for another 30 minutes.
In a medium-sized 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 Krapfen for 2-3 minutes per side, until they are golden brown. Use a slotted spoon to turn them.
Remove the fried Krapfen to a paper towel-lined plate to absorb excess oil.
Serve warm, dusted with powdered sugar. You can offer extra jam on the side, if desired.
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