Special equipment: a pastry bag fitted with a small round tip and a deep-frying thermometer
Bring a large pot of water to a boil and gently lower 10 of the eggs into the water. Set a timer for 12 minutes. Bring back to a boil, then reduce the heat to a vigorous simmer. When the timer goes off, drain and immediately place the eggs in a bowl of ice water. Let sit for 10 minutes. Peel the eggs, cut them in half and separate the whites from the yolks.
Transfer the yolks to a medium bowl. Add the mayonnaise, vinegar, mustard, 1/4 teaspoon salt and a few grinds of pepper. Mash with a whisk until combined, then whisk vigorously to smooth out any big lumps. Transfer the yolk mixture to a pastry bag fitted with a small round tip. Refrigerate until ready to use.
Fill a large heavy-bottomed pot with 2 inches of oil, attach a deep-frying thermometer and heat to 350 degrees F. Put the flour on a plate and season with 1/2 teaspoon salt and a few grinds of pepper. Crack the remaining 2 eggs into a shallow dish and beat lightly with 1/4 cup water. Put the panko on another plate. Line a third plate with paper towels.
Roll the egg whites first in the flour, then dip in the egg wash and coat in the panko. Fry in 2 batches, stirring occasionally, until golden brown, 2 to 3 minutes. Drain on the paper towel-lined plate round-side up and sprinkle with salt while hot.
Turn the egg whites over, fill with the yolk mixture and top with the paprika and jalapenos. (The fried eggs can be filled and served anywhere from warm to room temperature.)
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