Special equipment: a deep-frying thermometer
For the agliata sauce: In a medium bowl, toss the bread cubes with enough water to really moisten them up--you want it almost soggy. Squeeze out the excess water, and place the bread, vinegar and garlic in a food processor; puree until smooth. With the machine running, drizzle in the olive oil, and process until combined. Add the chives and parsley, and pulse a few more times to combine. Season with salt and more vinegar, if you like (I like a very bright, acidic dipping sauce). Transfer the sauce to a serving bowl.
For the Parmigiano-crusted cauliflower: Bring a large pot of well-salted water to a boil. Fill a large bowl of well-salted water with ice. Line a baking sheet with paper towels.
Toss the cauliflower in the boiling water. When the water returns to a boil, cook the cauliflower for another 2 minutes, then drain and immediately plunge the cauliflower into the ice water. When the cauliflower is cool, drain well and lay it out to dry on the prepared baking sheet.
In a large saucepan, pour enough oil to fill the pan to a depth of 1 1/2 to 2 inches. Heat over medium-high heat until a deep-frying thermometer registers 350 degrees F. To see if it's hot enough, drop some flour into the oil--if it sizzles and floats quickly, you're good to go; if the flour burns or the oil begins to smoke, it's too hot, so reduce the heat.
Place the flour in one bowl, the eggs beaten with 2 tablespoons water in a second bowl, and the breadcrumbs and Parmigiano in a third. Place a couple of layers of paper towels on a baking sheet next to the stovetop. Dredge some of the blanched cauliflower in the flour and shake off the excess, then dip it in the egg mixture, and finally in the breadcrumb mixture. Repeat for the remaining cauliflower.
Working in batches so you don't overcrowd the pan, fry the cauliflower until brown and crispy, 3 to 5 minutes. Transfer the cauliflower to the paper towels, sprinkle with salt, and serve hot, hot and hot with the agliata sauce.