Wearing plastic gloves, cut the stem end off of the peppers. Carefully remove the seeds and white membrane and stuff the peppers with cheese.
Heat the oil to 375°F in a deep fryer or large pot.
Create a breading station by placing the eggs in a small bowl and the bread crumbs in a shallow pan.
Drop 4 peppers into the eggs and toss to coat. Using a fork, lift 1 pepper at a time out of the eggs and shake off the excess. Drop the egg-coated pepper into the breadcrumbs and toss to coat. Place the breaded pepper on a sheet pan and repeat with the remaining peppers. When all the peppers are coated, let them sit to dry for 15 minutes.
With a slotted spoon, place the peppers, a few at a time, into the hot oil and fry for 2 to 3 minutes, until golden. Transfer the fried peppers to a platter lined with paper towels to drain. Repeat with the remaining peppers. Serve immediately.