1. Preheat grill to medium-high. Pluck parsley leaves and tender upper stems, discarding tough lower stems. Chop half of parsley (you should have about 1/2 cup packed). Gently fold half of chopped parsley into lamb just until evenly combined (do not overmix, or patties will be tough). Form into 4 patties slightly wider than buns, each about 1/2 inch thick. Season generously with salt and pepper; brush with oil.

2. Stir together cheese, yogurt, and remaining 1/4 cup chopped parsley; season with salt and pepper. In another bowl, stir together tomato, whole parsley leaves and tender stems, and oil; season with salt.

3. Brush grates with oil and grill patties, flipping once, until charred in places and cooked to desired doneness, 5 to 7 minutes for medium-rare; 9 to 11 minutes for medium. Transfer to a plate and let stand a few minutes. Meanwhile, grill split sides of buns just until charred in places, about 30 seconds.

4. Divide patties, yogurt-feta sauce, and tomato salad evenly among buns. Serve immediately, with fries.