Place an oven rack in the lower third of the oven and preheat to 425F. Make a slit, lengthwise, down the side of each pepper. Using a small spoon, scoop out the seeds and inner core of the peppers (if you like the peppers spicy you can leave some of the seeds in).
Coarsely shred 1 1/2 cups of cheese and cut the remaining into 16 small sticks. Place 1 cheese stick inside each pepper and set aside.
Crack the eggs into a shallow dish and season with salt (about 1 teaspoon) and pepper (about 1/2 teaspoon). Place the panko into a separate shallow dish and season with salt and pepper.
Dip each stuffed jalapeño in the egg, rolling it around to coat, and then the panko, pressing the breadcrumbs to each pepper. Transfer the breaded pepper to a 9-inch baking dish. Repeat with the remaining peppers.
Sprinkle the shredded cheese evenly over the top and roast in the oven until the peppers have softened and the cheese is browned and melted.
Top each pepper with salsa and serve.