In a large bowl, combine onion, yogurt, dill, hot sauce, salt, and minced garlic. Add ground turkey to this mixture and mix until smooth. Preheat grill to medium heat.
Form 4 cutlets, each roughly 1.5 cm thick. Place the cutlets on the heated grill rack and cook for 6-8 minutes on each side until the juices run clear and the internal temperature reaches 74°C (165°F).
Cut each burger bun in half. Layer lettuce, a cutlet, and a tomato slice between the two halves of each bun.