Prepare the dough: In the bowl of a stand mixer fitted with the dough hook, add the flour and crumbled fresh yeast. Turn the machine on at low speed. With the hook running, add the sugar and let it combine.
Add all of the lukewarm water, the vanilla extract, and the grated lemon zest, then let the mixer knead the mixture.
Add the egg and mix until fully incorporated. Increase the mixer speed slightly so the dough begins to develop structure. Pour in the olive oil in a slow, steady stream and let it absorb completely. Finally, add the salt and continue kneading.
Knead for about 20–25 minutes at medium speed. The finished dough should not be sticky at all; it should be smooth and elastic. Lightly oil your hands (do not use flour), shape the dough into a smooth ball, and place it into a lightly greased bowl. Cover with plastic wrap (or a shower cap) and let it rise in a warm place until doubled in size (about 3–4 hours).
Shaping the Doughnuts
Once doubled, gently deflate the dough and transfer it onto a work surface. You can shape the doughnuts in two different ways:
Method 1: Roll out the dough to a thickness of about 2 cm (approx. 3/4 inch). Using a round pastry cutter about 10 cm (4 inches) in diameter, cut out discs. Use a bottle cap or the base of a piping tip to cut a hole in the center of each disc to obtain uniform, perfectly round doughnuts.
Method 2 (Recommended for a rustic look): Divide the dough into 8 equal portions. Roll each piece into a rope, then pinch the two ends firmly together to form a ring.
Proofing and Frying
Cut out squares of parchment paper large enough to hold each doughnut. Place each shaped doughnut onto a parchment square, arrange them spaced apart on a baking sheet, and let them rise in a warm spot until doubled in size (about 1 hour).
Heat vegetable oil in a wide, deep pan over medium heat (the oil should be hot but not smoking, otherwise the doughnuts will burn on the outside before cooking through).
Carefully lower the doughnuts into the hot oil along with their parchment paper squares. As they fry, the paper will detach; gently remove it from the oil using kitchen tongs.
Fry for 2–3 minutes per side until golden brown. Remove with a slotted spoon and drain on paper towels.
While still warm, roll the doughnuts in granulated sugar to coat them completely, then transfer to a serving plate. Your lactose-free fried doughnuts are ready to serve!