How to Make It
Using clean hands, combine turkey, mayonnaise, oregano, cumin, salt, and 1/8 teaspoon pepper. Shape mixture into 4 equal patties.
Heat a large cast-iron skillet or grill pan over high. Lightly coat skillet with cooking spray, and add turkey patties. Cook until a meat thermometer inserted in thickest portion registers 165°F, 4 to 5 minutes per side.
Stir together yogurt, olives, lemon juice, and remaining 1/8 teaspoon pepper in a small bowl. Spread yogurt mixture evenly on cut sides of top and bottom buns. Divide arugula evenly among bottom halves of buns; top evenly with cooked patties, cucumber, and red onion. Cover with top halves of buns and serve.