Mixing feta into these burgers means you can use lean ground beef without sacrificing flavor or substance. The tangy yogurt and cucumber sauce is a refreshing alternative to other traditional toppings. Be sure to have plenty extra on hand, because your guests might really want to pile it on! The burgers can be cooked on a grill or in a pan on the stovetop.
To make the burgers: combine the beef, feta, oregano, red onion, half of the grated garlic, and black pepper in a large mixing bowl. Use your hands to gently combine the ingredients.
Form the beef mixture into four patties that are the same diameter as your rolls — the patties should be slightly thicker at the edges with a slight indentation in the middle. Let the burgers rest while you prepare the tzatziki.
To prepare the tzatziki: Peel your cucumber and grate it using the coarse side of a box grater (alternately, you can shred it using a vegetable peeler). In a small dish, combine the yogurt, cucumber, remaining garlic, and dill. Season to taste with salt. Cover and refrigerate until ready to serve.
Grill the burgers over a medium flame for approximately 5 minutes on each side (you can also cook them on the stovetop in a pan set over medium heat) or until they are cooked to your desired level of doneness.
Serve with rolls, lettuce, sliced red onion, and tzatziki.