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 2-inch lengths, adding them to the vinegar water as you work.
Drain the cardoon ribs, transfer them to a large saucepan of lightly salted water and boil until very tender, about 30 minutes.
Drain and pat dry; let cool.
In a shallow bowl, combine the egg yolks and egg. In another bowl, toss the crumbs with the Parmesan, 1/2 teaspoon of salt and 1/8 teaspoon of pepper. Dip the cardoon in the eggs, then dredge in the crumbs, pressing to help them adhere. Shake off any excess and set the cardoon on a wax paper-lined tray.
In a medium skillet, heat 1/2 inch of vegetable oil over moderately high heat. When the oil is very hot, add the cardoon, 5 or 6 pieces at a time, and fry until golden and crisp, about 1 1/2 minutes per side.
Transfer the cardoon to a wire rack set over a baking sheet.
Sprinkle lightly with salt and serve hot or at room temperature with lemon wedges.
