Preheat the air fryer to 370°F.
Open the 1 (16-ounce) jar of dill pickle coins and drain them. Use a clean kitchen towel or paper towels to pat the pickle coins dry on both sides; set them aside.
In a small bowl, whisk 1 egg until smooth. In a medium bowl, place 1 cup panko breadcrumbs.
Working one pickle at a time, dip a pickle coin into the beaten egg, letting excess egg drip back into the bowl, then press the pickle into the panko to coat both sides. Place each coated pickle in a single layer on a plate as you work.
Lightly drizzle extra-virgin olive oil over the coated pickles on the plate.
Arrange the coated, oiled pickles in a single layer in the air fryer basket with space around each piece. Do not overcrowd; work in additional batches if needed.
Air fry at 370°F for 11 to 14 minutes, flipping the pickles once halfway through cooking, until they are crisp and golden brown (exact time will vary by air fryer).
Transfer the fried pickles to a paper-towel-lined plate to drain briefly, then serve immediately with remoulade sauce for dipping.