Rub the chicken with 1 tablespoon of the olive oil and the oregano and season with salt and pepper.
In a large bowl, mix 2 tablespoons of the olive oil with the lemon juice and season with salt and pepper. Add the feta, tomatoes, cucumber, scallion, arugula and mint. Season the salad with salt and pepper and toss well.
In each of 2 large skillets, heat 1 tablespoon of olive oil until shimmering. Add the chicken cutlets and cook over high heat until golden on the bottom, about 5 minutes. Turn the chicken and cook just until done, about 2 minutes longer. Transfer the chicken to plates, spoon the salad on top or alongside and serve.