Preheat your air fryer to 400 degrees Fahrenheit.
In a shallow bowl, beat the egg. In another shallow bowl, combine the gluten-free breadcrumbs, salt, and pepper.
Dip the chicken cutlet in the egg, then coat it in the breadcrumb mixture.
Spray the air fryer basket with olive oil. Place the chicken cutlet in the basket and cook for 10-12 minutes, flipping halfway through, until cooked through and golden brown.
While the chicken is cooking, toss the arugula with the olive oil, salt, and pepper.
To serve, place the arugula on a plate and top with the cooked chicken cutlet. Sprinkle with feta cheese and serve immediately.