Start by preparing the burger patties. First, coarsely chop the mushrooms and add them to a food processor. Pulse several times until very finely chopped.
To a large bowl, add the chopped mushrooms, ground lamb, salt, and pepper. Mix well with your hands. Divide into 10 portions.
Take one portion of the lamb mixture and form a patty. Place some feta in the center. Take another portion of the lamb mixture, form another patty, and place it on top, pinching together the edges to seal. Repeat with the remaining lamb mixture and feta to make 5 stuffed burger patties.
Place the patties on a plate in the refrigerator to chill for at least 20 minutes.
Meanwhile, prepare the pickled red onions if you are planning to make them from scratch. Place your thinly sliced onion in a mason jar. In a small pot, combine the water, vinegar, honey, and salt. Bring to a boil, then turn off heat. Pour the vinegar mixture over the onions and let sit for ten minutes.
Also prepare the yogurt sauce. Combine the Greek yogurt, lemon juice, olive oil, salt, garlic powder, dill, and parsley in a bowl. Stir together and set aside for now.
When the burgers have chilled enough, remove the patties and place them in the basket of the air fryer. (Depending on the size of your air fryer, you may need to work in batches).
Air fry at 375 degrees F for 15-18 minutes, or until the lamb patties reach an internal temperature of 160 degrees F.
Place the lamb patties on a bun and top with arugula, a spoonful of pickled red onions, and the Greek yogurt sauce. Enjoy!