Place 1 finely chopped medium shallot, ¼ cup finely chopped fresh parsley leaves, ⅓ cup panko breadcrumbs, ¼ cup plain Greek yogurt, 1 tablespoon Dijon mustard, 1 teaspoon Worcestershire sauce, ½ teaspoon kosher salt, and ¼ teaspoon black pepper in a medium bowl. Mix to combine.
Add 1 pound lean ground turkey and mix until combined; do not overmix.
Divide the mixture into 4 (about 5-ounce) portions. Use your hands to gently flatten each portion into a patty about ½-inch thick and 3 ½ inches wide. Place on a plate or baking sheet. Press your thumb into the center of each patty to create a dimple, which will help keep the burgers from bulging in the center as they cook.
Heat 1 tablespoon olive oil in a large cast iron skillet or nonstick pan over medium heat until shimmering. Cooking in 2 batches if needed so as not to crowd the pan, add the patties. Cook undisturbed until a golden-brown crust forms on the bottom, about 6 minutes.
Flip the burgers with a thin, flat spatula. Top each burger with 1 slice cheese if desired. Cook until the burgers are cooked through and an instant-read thermometer inserted into the center of a patty registers at least 160°F, 5 to 6 minutes more.
Assemble the burgers on hamburger buns with desired toppings.
