Pour the excess moisture off the top of the yogurt and place it in a bowl. Grate the cucumber, then place the strips in a colander and toss with salt. Set aside for 20 minutes to drain, then squeeze out as much liquid as possible. Mix all ingredients in a bowl and salt and pepper to taste.
For the Hamburger Recipe:
Preheat grill to medium heat. Use your hands to mix the first 8 ingredients in a bowl. Gently fold in the feta cheese.
Divide the meat mixture into 6 equal portions, then shape it into 1-inch thick ovals. Brush patties with olive oil and salt and pepper.
Grill the patties for 10 minutes, carefully flipping once. Place the patties inside pitas and stuff with cucumbers, tomatoes, and mint leaves. Top with yogurt sauce.