Cut the fennel bulb in half lengthwise and then cut each half lengthwise into wedges that are 1/2 inch wide on the outside. You should get 12 to 16 wedges.
Bring about 2-1/2 quarts of water to a boil in a 4-quart saucepan over high heat. Add 2 Tbs. of the salt and the fennel. Boil briskly until the fennel is tender, 5 to 8 minutes. Check for doneness by removing a wedge to taste it. Drain well and set aside to cool.
In a shallow bowl, beat the eggs with the remaining 1/4 tsp. salt and several grinds of pepper.
In another shallow bowl, mix the breadcrumbs with the cheese.
Working with one or two wedges at a time, dip in the beaten egg, making sure the exterior is well coated. Lift out with a fork, letting the excess egg drain off. Then dredge in the breadcrumbs, patting the breadcrumbs in place so they adhere (you want to coat them well). Keep the wedges compact; don’t let them splay open. Set the wedges on a tray and continue until all are coated.
Tip: If you don’t have a thermometer or your thermometer won’t reach far enough into the oil to read accurately, you can test the oil temperature by adding a few breadcrumbs. If they sizzle immediately and float to the top, the oil is ready.
Put 1/2 inch of oil in a 10-inch straight-sided sauté pan, attach a candy thermometer to the side of the pan, and heat over medium-high heat. When the oil reaches 375ºF, add as many wedges as will fit comfortably in a single layer. Don’t crowd the pan. Cook until well browned on both sides, turning once with tongs. Total frying time should be about 1 minute. Transfer the wedges to a plate lined with paper towels and sprinkle lightly with salt. Continue frying, adding more oil to the pan as needed, until all the wedges are fried. Serve hot.