Start by making the tzatziki sauce. In a medium bowl, combine Greek yogurt, grated cucumber (squeezed of excess moisture), fresh dill, and lemon juice. Season with salt and black pepper to taste. Stir well, cover, and refrigerate while preparing the burgers.
In a large bowl, add ground turkey, breadcrumbs, finely chopped onion, minced garlic, dried oregano, salt, and black pepper. Gently mix until just combined, being careful not to overmix.
Divide the turkey mixture into 4 equal portions and shape each into a round patty about 3/4-inch thick. Press thumb lightly into the center of each patty to prevent puffing during cooking.
Heat olive oil in a large skillet over medium heat. Cook the patties for about 5-6 minutes per side, until internal temperature reaches 165°F (74°C) and patties are golden brown. Avoid pressing down on patties.
Lightly toast burger buns if desired. On each bun bottom, layer a lettuce leaf and a turkey burger patty. Top with a generous dollop of tzatziki sauce and tomato slices. Place bun top on and serve.