Using a long-handled fork, char chiles over an open flame, turning, until skins are blackened, 4 to 6 minutes. (Alternatively, broil chiles on a broiler pan about 2 inches from heat, turning occasionally until skins are blistered and charred.) Transfer chiles to a bowl and let stand, covered, until cool enough to handle. Using a small knife, carefully scrape the charred skin off the chiles (leave the stems attached).
Carefully cut a lengthwise slit in each chile to form a pocket, and cut out and discard the seeds. Stuff each chile with some of the mild cheddar, and thread shut with a toothpick. In a large saucepan, melt enough shortening to reach 1/2-inch up the sides, and heat to 375 degrees F. on a deep-fat thermometer.
Preheat the broiler. In a bowl, combine flour, salt, and pepper. In another bowl, carefully fold yolks into egg whites. Working in batches, dredge the chiles in the flour and, using a slotted spoon, dip into the eggs to coat completely. Fry, turning occasionally, until golden, no more than a couple of minutes. Using a slotted spoon, transfer chiles (careful, they are delicate) to paper towels to drain.
Arrange chiles on a flameproof plate and sprinkle with the sharp cheddar. Broil chiles just long enough to melt cheese and serve immediately, accompanied by salsa, if desired.