Start by cooking the spinach. You can sauté it in a little olive oil until wilted or steam it. Once cooked, allow it to cool slightly before chopping it finely.
In a large mixing bowl, combine the ground turkey, cooked spinach, garlic powder, kosher salt, dry oregano, and feta cheese. Use your hands to mix everything together until well combined, but be careful not to overmix, as this can make the burgers tough.
Divide the mixture into four equal portions and shape them into patties. Make a small indentation in the center of each patty with your thumb; this helps them cook evenly.
Heat a skillet or grill over medium heat and add the tablespoon of olive oil. Once hot, place the patties on the skillet or grill. Cook for about 5-6 minutes on each side, or until the internal temperature reaches 165°F.
While the burgers are cooking, prepare the tzatziki sauce. In a mixing bowl, combine the Greek yogurt, minced garlic, shredded cucumber, dry dill, olive oil, lemon juice, and kosher salt. Mix well and set aside to let the flavors meld.
Toast the whole-wheat burger buns lightly on the grill or in a toaster. Spread a generous amount of tzatziki sauce on the bottom half of each bun. Top with a turkey patty, then add slices of tomato, red onion, and any additional toppings you desire. Finish it off with the top half of the bun.