Add the flour, yeast, sugar, and salt to a large bowl and mix well. Then add the water and combine. Transfer to a floured work surface and knead for about five minutes until smooth. Lightly oil a large bowl, shape the dough into a ball, and coat with the olive oil that is in the bowl. Cover bowl with plastic wrap. Let dough stand for about two hours until doubled in size. Once doubled in size, cut the dough in half. Using a rolling pin on a floured work surface, roll one piece out into a sheet about six millimeters thick. Using a pasta cutter or pastry wheel, cut into small squares, rectangles, or diamonds, about five centimeters wide. Repeat with remaining dough.
Heat oil in a large, deep skillet over medium heat to 350F. Start to fry, adding a few at a time and turning as they start to turn golden. Remove with a slotted spoon and drain in a bowl on paper towels. Season each batch with plenty of salt and serve.
For the broad bean dip:
Bring a saucepan of water to a boil and cook the beans for five minutes. Drain and, when cool enough to handle, remove the skins.
Put the beans in a food processor with the lemon juice, garlic, mint, pecorino, and extra-virgin olive oil. Season with salt and pepper and blend to form a coarse paste. Serve in a small bowl for dipping.