Preheat the oven to 350 degrees F. Generously coat a 13-by-9-inch straight-sided baking dish (preferably glass) with cooking spray.
Separate the egg whites and yolks into 2 medium bowls. Whisk the whites until combined and just foamy, then transfer them to the prepared baking dish. Cover the dish with foil and bake until they have fully set, about 20 minutes. Let cool completely in the baking dish.
Meanwhile, beat the yolks until smooth. Lightly coat a medium nonstick skillet with cooking spray. Add the yolks, and scramble over medium-low heat until fully cooked and slightly dry, 4 to 5 minutes.
Transfer the yolks to a food processor. Add the sour cream, mayonnaise, vinegar, mustard, hot sauce, 1/2 teaspoon salt and a few grinds of pepper, and pulse until smooth and creamy. Spoon the yolk mixture into a pastry bag fitted with a small round tip, or a resealable plastic bag with the tip cut off.
Put a platter on top of the baking dish with the egg whites and flip to turn them out onto the platter. With a long sharp knife, trim about 1/4-inch from the edges, including any browned outer edges, to make a neat rectangle of whites. Cut the trimmings into 1/4-inch cubes that will be used for "stars" (discard any with browned edges). Cut the large rectangle of egg whites into 54 square pieces: 6 rows across and 9 columns down. Straighten the eggs to keep the rectangle together. Pipe a heaping amount of the egg yolk mixture onto the upper left-hand corner of the rectangle (this will be the section of the flag that holds the stars). Use the rest of the yolk filling to generously pipe 6 horizontal stripes right over the middle of each egg (see Cook's Note). Sprinkle the corner square with parsley and the stripes with paprika or cayenne. Top the parsley with the small dice "stars".
There will be some yolk mixture remaining. We think it's delicious on toasted bread or crackers with a sprinkle of cayenne or paprika.
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