In a bowl, combine the water and vinegar. Using a vegetable peeler, remove the tough outer strings of the cardoon ribs.
Cut the ribs into two inch lengths, placing them in the water as you work. Drain the cardoon ribs and place them into a large saucepan of lightly salted water and boil until tender, about 30 min. Drain and pat dry, let cool. In a shallow bowl combine the egg yolks and egg. In another bowl, toss the bread crumbs with the parmesan cheese, 1/2 teaspoon of salt and pepper. Dip the cardoon in the eggs and then dredge in the parmesan and crumb mixture, patting if necessary to adhere. In a medium skillet, heat 1/2 inch of vegetable oil over moderately high heat. When the oil is very hot, add the cardoon, five or six pieces at a time, and fry until golden brown, about 1 1/2 minutes per side. Sprinkle lightly with salt and pepper and serve hot with lemon wedges.