Pat the peppers dry with a paper towel and set aside.
Wrap each cheese stick with a slice of ham, then slice the wrapped sticks into approximately 1/2-inch cubes. Stuff the wrapped cheese cubes into the peppers.
Heat the oil in a medium heavy-bottomed pot to 350 degrees F.
Set up a breading station using pie pans or plates. Put the flour in the first pie pan, then whisk together the egg and 2 tablespoons water in the second pie pan. In the third, mix together the panko, salt and pepper. Run each stuffed pepper through the breading station: lightly coat in the flour, dip in the egg wash and thoroughly coat in the panko.
Fry the peppers in batches in the hot oil until the breading is lightly browned, 1 to 2 minutes. Transfer to a plate lined with a paper towel. Serve hot.