In a small bowl, stir together the yogurt, oil, lemon juice, garlic, dill, and salt and pepper.
Heat 2 teaspoons of oil in a nonstick skillet over medium-high heat. Add the onion and cook until soft and translucent, about 3 minutes. Add the spinach and cook until wilted, about 1 minute. Remove the pan from the heat. Add the feta cheese, dill and 1/4 teaspoon black pepper and stir to combine.
Divide the turkey into 4 equal sized rounds. Make 2 equal sized patties out of each round so you have 8 patties total. Put 2 tablespoons of the spinach-feta mixture onto half of the patties. Top with remaining patties working the turkey around the edges to seal burgers closed. Season the burgers on both sides with the salt and remaining 1/4 teaspoon pepper.
Spray a nonstick grill pan with cooking spray and heat over medium-high heat, or prepare the grill. Grill the patties until cooked through, about 5 minutes per side.
To serve, place a burger on the bottom half of each bun, top with about 2 tablespoons of yogurt sauce, then 2 or 3 cucumber slices and a lettuce leaf. Top with the other half of the bun and serve.