Heat olive oil in skillet over medium heat. Add bell pepper, onion, and garlic and cook until onions are caramelized.
Remove from heat and add to food processor (or blender) with 1/2 cup olive oil mayo. Pulse until there are a few bits of pepper and onion remain, but mostly smooth.
Combine ground turkey with one egg, chopped spinach, feta cheese, and Greek seasoning. Form into four patties.
Heat olive oil in skillet over medium heat. Add patties and cook until browned on each side, about 5 minutes on each side. Once they are browned and firm on the outside, but juicy inside, they are ready. Set patties aside.
Add buns to skillet, and lightly toast on each side. To serve, spread aioli on buns and burger. Top with lettuce, additional cheese, onions, and kalamata olives, if desired.