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Dissolve the yeast with 1/3 cup of the milk in a bowl. In a separate bowl, whisk the eggs with the remaining milk. In the bowl of a stand mixer fitted with a dough hook, add the flour, sugar, salt and lemon zest, and mix together.
Add the egg and milk mixture slowly to the flour, and mix until the dough is roughly combined, 2 to 3 minutes. Add in the yeast mixture, and continue to mix the dough for a few minutes. Add the butter to the dough a little at a time until mixed in. Sprinkle in a little flour if the dough is sticking to the bowl and mix until it no longer sticks to the sides. The dough should be very soft and not sticky. Transfer the dough onto a work surface and knead the dough into a ball, then put in a bowl and cover with plastic wrap. Proof the dough until doubled in size, 3 hours.
Roll out the dough with a rolling pin until about 3/4-inch thick. Using a round pastry cutter the size of a cup, cut the dough into disks. Place the dough on a baking tray lined with parchment paper and cover with plastic wrap. Proof the dough for another 1 hour 30 minutes.
Add the pieces of dough, a few at a time, to the hot oil. When the dough is golden on one side gently turn over to fry the other side, 3 to 5 minutes. When the doughnuts are golden remove from the fryer and place on paper towels to absorb the excess oil. Then roll the doughnuts in sugar until completely covered.
Fill a pastry bag fitted with a pastry tip with the chocolate hazelnut spread. Poke the tip three-quarters into the doughnut and squeeze the bag, pulling the tip out and adding as much spread as possible to the doughnut. Serve warm.