Shred or grate the cucumbers and blot them with a paper towel or clean kitchen towel to get rid of as much moisture as possible.
Mix the shredded cucumbers, strained yogurt, garlic, vinegar, and lemon juice.
Add dill and salt and pepper to taste. Cover and refrigerate for 1 hour.
In a large bowl, stir together ground turkey, egg, feta cheese, red onion, oregano, lemon zest, and salt. Shape mixture into 8 (1/2-inch-thick) patties.
Heat the griddle to medium-high heat and cover with a thin layer of olive oil.
Add patties to the hot griddle; cook approximately 5 minutes on each side or until done.
Serve each patty on a hamburger bun with homemade tzatziki sauce and desired burger toppings.
