Heat a gas or charcoal grill to medium-high (400°), or heat a stovetop grill pan over medium-high heat.
Mince garlic with salt, then smash with the flat side of the knife into a paste. Scrape into a large bowl and stir in ginger, onion, cayenne, coriander, cumin, allspice, pepper, and parsley until thoroughly combined.
Add ground beef. With your hands, gently mix.
Shape mixture into 6 oblong patties that will each fit in a pita half, making patties about 3/4 in. thick. Oil cooking grate, using tongs and a wad of oiled paper towels. Grill patties until well browned and just pink inside, 4 to 5 minutes per side.
Cut pita breads in half crosswise. Gently loosen the halves to form pockets. Brush each with oil and add to grill. Toast pita until pale golden and crisp, 1 to 2 minutes. Spread inside of each pocket with 1 to 2 tbsp. tzatziki. Tuck burger inside, followed by cucumber, tomato, lettuce, and mint leaves.