Grate 1/2 an unpeeled English cucumber using the large side of a box grater and squeeze out the water using a cheesecloth or your hands. To a mixing bowl, add Greek yogurt, minced garlic, grated cucumber, olive oil, red wine vinegar, salt, and pepper. Store in an airtight container in the fridge.
In a separate bowl, combine ground beef, red onion, garlic, feta, fresh dill, dried oregano, salt, and pepper. Roll into 4 balls.
Heat a flat top grill or cast-iron pan over medium-high heat and coat with a light layer of olive oil.
Transfer meat to the grill and place a pita on top. Immediately smash down using a burger press (or heavy pan), until it’s as flat as you can make it.
Cook for 4 minutes until the meat is cooked through. Flip over for about 30-60 seconds. Transfer cooked pita burgers to a wire rack or paper towel lined plate to remove excess oil.
Top each pita burger with desired amount of tzatziki, tomatoes, cucumber, red onion, and crumbled feta.