How to Make It
Preheat a grill to high (450°F to 550°F), or heat a grill pan over high.
Combine beef, lamb, feta, onion, lemon zest, lemon juice, cumin, paprika, pepper, and 1/4 teaspoon salt in a bowl; stir well (or use your hands) to incorporate. Divide mixture into 4 equal portions; shape portions into oval patties. Lightly coat grill grate with cooking spray; place patties on oiled grate. Grill to desired degree of doneness, 2 to 3 minutes per side for medium-well.
Place shredded cucumber on paper towels to drain 5 minutes. Stir together drained cucumber, yogurt, and remaining 1/4 teaspoon salt in a bowl.
Place pita halves on grill grates; grill until lightly toasted, about 1 minute per side. Spoon about 2 tablespoons yogurt sauce into each toasted pita. Divide patties, tomato slices, and lettuce evenly among each pita half.