Slice the onion and sauté in olive oil over medium heat until dark in color, add fresh thyme and remove from heat. Set aside and let cool. Meanwhile, mix kalamata olives with mayonnaise. Set aside. Mix oil and vinegar together in a small bowl. Set that aside also. Add sun-dried tomatoes to onion mixture, then add turkey. Mix and combine into four burgers. Using your finger, make an indentation in each patty. Insert a cheese cube in each burger and re-form them so that the cheese is inside. Brush each burger with olive oil and vinegar on one side. Then salt and pepper. Place the seasoned side down in a pre-heated grill pan. Grill for 6 to 7 minutes or until the meat reaches 160/165°F. Before turning over on the second side, brush with the oil and vinegar and sprinkle with salt and pepper. Remove from the pan and let rest for a couple of minutes. Layer the kalamata mayonnaise, lettuce, tomato and burgers on buns.