Using a large hole box grater, shred zucchini; transfer to a clean kitchen towel.
Gather corners of towel and squeeze tightly to release excess liquid. (When you think you’re done, squeeze a bit more.) Shredded zucchini should measure about 2 cups (8 ounces in weight).
Combine 1 cup shredded and squeezed zucchini, turkey, ½ cup yogurt, bread crumbs, parsley, oil, oregano, Worcestershire, 1 ½ teaspoons garlic, ½ teaspoon salt, and ¼ teaspoon red pepper flakes.
Divide meat into 4 equal portions.
Gently form each into a ball, then lightly flatten into 1-inch-thick burger patties. Press finger through center of patties (this helps keep the patties flat during cooking) and transfer to a plate; chill patties at least 15 minutes or up to 4 hours.
For the sauce, combine remaining 1 cup shredded and squeezed zucchini, remaining 1 cup yogurt, remaining ½ teaspoon garlic, dill, vinegar, ½ teaspoon salt and ¼ teaspoon black pepper; season with additional salt and pepper to taste.