Roast chilis on their sides over a gas flame over medium-heat, turning with tongs, until skins are blistered and slightly charred, 4 - 6 minutes. (Or, broil on a rack about 2 inches from heat).
Transfer to a large bowl and cover with plastic wrap, let stand for 20 minutes. (This can be done ahead of time).
Carefully rub off skins from chilies, leaving stems attached.
Cut a slit lengthwise in each chile using kitchen shears or a sharp knife.
Carefully cut out ribs and the seedpod. Clean out all seeds with a wet paper towel, if necessary (Leaving the seeds will make the peppers more hot).
Stuff chilies with cheese and enclose filling by overlapping slit slightly, then transfer to a plate. (All this can be done the day before).
Heat 1/2-inch of oil in another large heavy skillet over medium heat until it registers 350 F on thermometer.
While oil heats, beat egg whites with 1/8 teaspoon salt, using an electric mixer until they just hold stiff peaks.
Whisk yolks with another 1/8 teaspoon salt in a large bowl, then gently fold in whites.
Generously dust chilies with flour, coating all sides.
When oil is ready, dip chilies, 1 at a time into egg batter, spooning it over chile to cover completely.
Fry the chilis, turning once, until golden on all sides (Lap oil over any lighter areas, 4 to 5 minutes).
Transfer chilies with a slotted spoon to paper towels to drain.
Meanwhile, re-heat sauce.
Ladle sauce onto plates and place a chili on top of sauce.