Special equipment: a deep-fry thermometer and a pastry piping bag with a 1/4-inch round tip
For the pizza dough: Add the warm water in a small bowl and sprinkle over 1 teaspoon of the sugar and the yeast; set aside so the yeast can proof.
Oil two bowls with a little olive oil and set aside.
In the bowl of a stand mixer fitted with a dough attachment, mix together the olive oil, salt, 3 1/2 cups of the flour and the remaining 1 1/2 teaspoons sugar. Stir the yeast in the water to make sure it's all dissolved and add it to the flour mixture. Mix on medium-low speed until everything is combined. Increase the speed to medium and allow the machine to knead the dough, adding more flour if necessary, until it is smooth and comes away from the side of the bowl, 6 to 7 minutes.
Divide the dough in two pieces and roll each into a ball. Place a ball in each of the oiled bowls seam-side down, and brush the tops with a little oil to stop them from drying out. Cover the bowls with plastic wrap and place in a warm place (inside a microwave or a turned-off oven works best) until it doubles in size and when poked with a finger, the indent remains (it doesn't spring back), about 1 1/2 hours.
For the zeppole: Line a baking sheet with parchment paper and sprinkle lightly with some flour; set aside. Place some granulated sugar in a shallow bowl.
Cut 1 ball of pizza dough into a grid pattern so you end up with 25 semi-equal little squares of dough. (Reserve the remaining ball for another use; store in the refrigerator for up to 24 hours.) Place them on your prepared baking sheet, cover them with a lint-free towel and allow them to rest somewhere warm for about 20 minutes.
Pour enough vegetable oil in a large pot to come up two inches and heat to 375 degrees F on a deep-fry thermometer. Fry the dough squares in batches until golden brown all around, 2 to 3 minutes.
Drain the zeppole on a paper-towel lined plate, then toss in the granulated sugar to coat. Transfer to a plate and allow to sit for 10 minutes.
Place the chocolate-hazelnut spread in a piping bag fitted with a plain round tip (about 1/4-inch round). Using a skewer, poke a small hole on one side of a zeppole and fill with the chocolate hazelnut spread. Repeat with the remaining zeppole. These are best eaten when slightly warm.
You can use store-bought pizza dough instead of homemade.