In a large mixing bowl, combine the ground turkey, gluten-free breadcrumbs, grated Parmesan cheese, chopped red onion, minced garlic, egg, dried oregano, salt, and black pepper.
Gently mix the ingredients until they are well combined. Be careful not to overmix, as this can make the burgers tough.
Divide the mixture into 4 equal portions and shape them into burger patties. You can make them as thick or thin as you prefer.
Heat a grill or a stovetop pan over medium-high heat and grease it lightly with cooking spray or oil.
Place the turkey burger patties on the grill or in the pan and cook for about 4-5 minutes per side, or until they are cooked through and have an internal temperature of 165°F (74°C).
While the burgers are cooking, you can toast the gluten-free hamburger buns, if you'd like.
Serve the gluten-free turkey burgers on the buns, and add your favorite gluten-free toppings, such as lettuce, tomato, avocado, cheese, or condiments.