Place the turkey, egg yolk, oregano, salt and pepper in a large bowl. Grate half the red onion and add it to the bowl. Mix with your hands, then form into four 1/2-inch-thick patties. Place on a sheet pan or plate lined with parchment paper.
Put the pitas on a baking sheet and place in the oven to warm.
Heat the olive oil in a large cast-iron or other nonstick skillet over medium-high heat. When the oil is hot, add the patties and cook until they are brown on the outside and no longer pink in the center, approximately 5 minutes on each side.
Remove the pitas from the oven and cut the tops off. Slice the remaining 1/2 red onion. Spread some tzatziki inside each pita. Fill each with lettuce, a patty, some chopped cucumber, red onion and feta, then serve with chips on the side.