Place 6 eggs in an air fryer basket. Cook at 250° for 17 minutes. Immediately (and carefully!) transfer eggs to an ice bath. Let sit until cool enough to handle, then peel.

Slice eggs in half lengthwise. Scoop out yolks and transfer to a medium bowl; set egg whites aside. To yolks, add mayonnaise, relish, 1 tsp. Dijon mustard, and 1 tsp. yellow mustard. Mash and stir with a fork until very smooth and creamy; season with salt and black pepper. Transfer to a small piping bag or gallon zip-top bag (we fit our piping bag with a small open star tip; feel free to use any small piping tip or snip the corner off the bag). Refrigerate until ready to use.

Pour flour into a shallow bowl. In another shallow bowl, lightly beat remaining 2 eggs, ½ tsp. Dijon mustard, and ½ tsp. yellow mustard; season with a pinch of salt and black pepper. In a third shallow bowl, combine cracker crumbs, panko, garlic powder, parsley, cayenne (if using), ¾ tsp. salt, and ¼ tsp. black pepper.

Working one egg white shell at a time, dip into flour, then into egg mixture, letting excess drip off. Generously coat in panko mixture, gently pressing to adhere. Arrange on a baking sheet. Spritz both sides of each egg with a light coating of cooking spray.

Working in batches if needed, arrange eggs in air fryer basket without overcrowding. Cook at 390°, turning eggs halfway through, until deeply golden and crispy, 8 to 10 minutes. Let cool slightly.

Arrange egg whites on a platter. Pipe yolk mixture into egg whites. Dust with paprika. Top with dill.