Fried Cardoons

Discard any discolored outer stalks and small leaves from cardoons. Trim base, tips, and outermost stalks, removing strings from stalks with a vegetable peeler (as for celery), then cut cardoons crosswise into 2-inch pieces.
Soak cardoons in 2 quarts water with 1 tablespoon salt in a large bowl, chilled, at least 8 hours.
Drain cardoons. Transfer to a 6-quart pot, then generously cover with cold water and add 2 tablespoons salt. Bring to a boil, then simmer, uncovered, until tender when pierced with a sharp knife, 20 to 25 minutes. Drain on several layers of paper towels and cool 5 minutes.
Heat about 1 1/2 inches oil to 360°F in a 4- to 5-quart heavy pot over medium heat.
Whisk together flour and 1/4 teaspoon pepper in a shallow bowl. Whisk together eggs, water, and cheese in another shallow bowl.
Toss one fifth of cardoons (all at once) with flour, shaking off excess, then transfer to egg mixture and turn with a fork to coat. Lift out coated cardoons 2 at a time, letting excess drip off, then carefully drop into hot oil. When all of batch has been added, fry, turning occasionally, until golden, 2 to 4 minutes. Transfer with a slotted spoon to fresh paper towels to drain.
Coat and fry remaining cardoons in 4 batches. (Return oil to 360°F between batches.) Season with salt.