Mound 3 1/2 cups of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs and 1/2 teaspoon olive oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour, starting with the inner rim of the well.
As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. The dough will come together when half of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up and discard any leftover bits. Lightly re-flour the board and continue kneading for six more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature.
Roll the pasta out to the thinnest setting on a pasta rolling machine. Using a pasta cutter or very sharp knife, cut the past into ribbon 1/8-inch thick. Set aside on a tray dusted with semolina flour, and cover with a clean, damp dish towel until ready to cook.
Bring 6 quarts of water to a boil and add 2 tablespoons salt.
In a large, non-reactive bowl, combine all vegetables and toss with the lemon juice, 3 tablespoons olive oil and salt and pepper, to taste.
Drop the pasta into the boiling water and cook 1 minute. Drain and add to the bowl with the dressed vegetables. Add the basil leaves and toss to coat. Divide among 6 warmed pasta bowls and serve topped with grated Parmigiano-Reggiano.