Pour the flour into a stand mixer, add the water mixed with the fresh yeast, then add the olive oil and salt. Knead for 10 minutes, cover the bowl with plastic wrap, and let the dough rise for 1 hour and 30 minutes.
Once the rising time has passed, take the dough, cut it with the help of a knife, roll it out, and form small discs. Lightly dredge them in flour again.
Fry the discs a couple at a time in a pan with plenty of hot peanut oil for a few minutes until golden brown. Drain them and transfer to paper towels to remove excess oil.
Finally, top each fried pizzetta with a drizzle of tomato purée, shaved or grated Parmigiano Reggiano (or diced fiordilatte mozzarella), and a small basil leaf. Serve immediately.
