Combine the yogurt, feta, cucumber, mint, half of the garlic, 1 tablespoon water, 1/2 teaspoon salt and a few grinds of pepper in a small bowl. Refrigerate until ready to serve.
Combine the turkey, the remaining garlic, 1 teaspoon salt and a few grinds of pepper in a medium bowl. Form into four 1/2-inch-thick patties. Heat a large nonstick skillet over medium-high heat. Add the olive oil, then add the patties and cook until browned and cooked through, 4 to 5 minutes per side. Transfer to a plate.
Spread the yogurt sauce on the buns. Sandwich with the burgers; top with lettuce and tomato. Serve with potato chips.