Peel and halve the hard-boiled eggs lengthwise, fork-mash their yolks in a medium bowl, then mix in the mayonnaise, mustard, pickle juice or vinegar, pepper, 1/4 teaspoon salt, and optional hot sauce before transferring the filling to a zip-top bag with a 1/4-inch opening cut from one bottom corner. The mixture should look mostly smooth and hold together rather than flow like a sauce.

Bring a 1 1/2-inch depth of vegetable oil to 350°F in a large Dutch oven over medium-high heat, checking it with a deep-fry thermometer. A steady thermometer reading signals that the oil is ready for breading to brown promptly.

Arrange three shallow bowls containing flour mixed with the remaining 1/2 teaspoon salt, the 2 raw eggs whisked together, and panko mixed with parmesan, then cover each dry white with flour, beaten egg, and crumbs in that order, letting excess egg drain and gently pressing the final coating into place. The curved surface should have an even crumb covering, with the center hollow still open enough to hold filling.

Lower the coated whites into the oil in batches, fry for 1 to 2 minutes, then lift them onto a cooling rack, sprinkle with salt, and allow them to cool slightly. Look for a golden crust and wait until the halves feel barely warm before filling them.

Pipe a generous mound of the yolk mixture into each fried white and finish with paprika and chopped chives. The filling should sit neatly inside the hollow with the crunchy rim still visible.