Tzatziki: Stir the yogurt with the cucumber, vinegar, mint, dill, oil, salt and garlic. Chill for at least 2 hours.
Sliders: Preheat the grill to medium-high; lightly coat the grate with cooking oil. Stir the breadcrumbs with 1/2 cup tzatziki, feta, lemon zest, oregano, salt, pepper, garlic and egg. Crumble in the ground chicken and gently mix to combine. Form into 16 patties. Grill, turning as needed, for 15 minutes or until well-marked and cooked through. Split the pitas open. Spread some tzatziki on the inside of the pita. Add lettuce, tomato, cucumber and a cooked pattie.
Prepare the sliders in advance and pack along all the garnishes separate when going on a picnic or to the beach for the day. Slider can also be cooked in a preheat 400°F oven for 15 to 20 minutes.