It may be difficult to find suitable slider buns. Pepperidge Farm makes one that I find serviceable, though it could be better; some packaged dinner rolls are acceptable. For this recipe, pita works fine. Of course, it could be used just as well for larger hamburgers.
Mince the garlic with the salt. Blend with the meat in a bowl along with the onion, parsley, cumin, coriander, cayenne, ginger, and pepper. Refrigerate for at least thirty minutes.
Greek yogurt is often thick enough for a tzatziki, but if yours is not, drain it in a sieve though a paper towel for an hour. (Domestic yogurt may be substituted, but generally requires more draining time.) Mix the yogurt with the lemon juice, two tablespoons of olive oil, and salt to taste in a small bowl and refrigerate.
Peel, seed, thinly slice, and chop the cucumber. Slice and lightly salt the tomato. Chop the mint. Divide the meat into balls of the desired quantity and flatten into patties. Spread the tzatziki on both sides of each bun. Sauté the patties on both sides in olive oil and serve on buns, covered with cucumber, tomato, and mint.