Toss the cucumber, feta, vinegar, mint, 1 tablespoon olive oil and a pinch each of salt and pepper in a bowl; cover and refrigerate.
Mix the turkey, 1/2 cup hummus, the parsley and coriander in a bowl; season generously with pepper. Dampen your hands and shape the mixture into 16 small patties, about 1/2 inch thick.
Heat 1 1/2 tablespoons olive oil in a medium cast-iron skillet over medium-high heat. Add half of the patties and cook until browned and cooked through, about 2 minutes per side. Transfer to a plate. Cook the remaining patties in the remaining 1 1/2 tablespoons olive oil.
Mix the remaining 1/2 cup hummus with a splash of hot water in a bowl. Spread some of the hummus on the inside of each pita; fill with a tomato slice, turkey patty and some of the cucumber mixture.