Spray both sides of the patties generously with oil and transfer them to the air fryer basket in one layer (you will probably have to cook these burgers in batches depending on the size of your air fryer). Air-fry each batch at 400ºF for 10 minutes, flipping the burgers over halfway through the cooking time.
While the burgers are cooking, make the cucumber yogurt dill sauce by mixing all the ingredients in a bowl.
Build your burger on the whole-wheat hamburger buns with arugula, tomato and the cucumber yogurt dill sauce.