Step 1

Preheat grill to medium-high or preheat broiler to high.

Step 2

Mix yogurt, feta, 1 tablespoon oregano, lemon zest, lemon juice and 1/4 teaspoon salt in a small bowl.

Step 3

Cut 1/4-inch-thick slices of onion to make 1/4 cup. Finely chop more onion to make 1/4 cup. (Reserve any remaining onion for another use.) Mix the chopped onion and meat in a large bowl with the remaining 2 tablespoons oregano and 1/2 teaspoon each salt and pepper. Form into 4 oval patties, about 4 inches by 3 inches.

Step 4

Grill or broil the burgers, turning once, until an instant-read thermometer registers 160 degrees F, 4 to 6 minutes per side. Serve in pita halves, with the sauce, onion slices, cucumber and tomato.