Peel shrimp but leave the tails on. (Use a small paring knife to make a tiny cut just above the tail to make it easier to start peeling.) Remove the inner and outer black veins (really the digestive tract -- gross) by loosening them with the knife or the pointy tip of a metal skewer, then pulling them out in one piece with your fingers. Discard the veins. Rinse the shrimp in cold water and pat dry with paper towels.
Wash your hands well and beat the eggs in a small bowl. Add salt and pepper. Place the bread crumbs in a dinner plate. Set up an assembly line from left to right with the shrimp, the eggs, the bread crumbs, and a sheet pan with sides.
Grasping the tail of the shrimp, dip each one in the eggs, then dredge in the bread crumbs, patting the crumbs onto the shrimp. Line up the shrimp on the sheet pan, using wax paper to separate the layers of shrimp if you need to stack them.
Wash your hands again and set a clean sheet pan or platter lined with paper towels near the stove, along with tongs (or a fork) and a slotted spoon. Pour enough oil in your frying pan to cover the bottom. Heat the oil on medium high. When the oil is hot enough to sizzle, add a layer of shrimp at a time, without letting them touch. Fry on the first side until the bottom is browned, then carefully flip over the shrimp, using tongs or a fork. When the shrimp is browned on the second side (it should take up to 5 minutes total for the shrimp to cook) remove the shrimp with a slotted spoon and place it on the paper towel-lined pan to drain.
Continue frying all of the shrimp in batches. After your second batch, discard the oil in the pan (place it in a disposable container rather than dumping it down your drain to prevent problems) and replenish the pan with fresh oil, enough to cover the bottom of the pan. (It's necessary to change the oil after every couple of batches, because once it gets dark, it will burn the shrimp on the outside without cooking it all the way.)
Blot the cooked shrimp and serve with the lemon cut in half or wedges so it can be squeezed on top, and cocktail sauce on the side. Store leftovers in the refrigerator for up to three days.
This recipe is for a large batch (3 pounds), so the preparation time will be much less if making a smaller batch. If you are making a different amount, use one jumbo egg per pound of shrimp.