Make the Greek yogurt sauce: in a small bowl combine 3/4 cup plain Greek yogurt, 1 tablespoon lemon juice, 2 grated garlic cloves, 1 tablespoon finely minced parsley, 1 tablespoon finely minced mint, and 1/2 teaspoon kosher salt. Stir until smooth, taste, adjust seasoning, then cover and refrigerate until ready to serve.
Prepare the burger mixture: in a large bowl add 1 pound ground chicken, 1/4 cup crumbled feta, 1/4 cup finely diced red bell pepper, 1 tablespoon finely minced mint, 1 tablespoon finely minced parsley, 1/4 teaspoon kosher salt, and 1/4 teaspoon ground black pepper.
Combine gently with your hands until evenly mixed, being careful not to overwork the meat.
Form the mixture into 4 equal-sized patties and place them on a plate or baking sheet.
To cook in a skillet: heat a 12-inch skillet over medium-high and add 2 tablespoons olive oil. When hot, add the patties and cook until golden and cooked through, about 5 minutes per side, or until internal temperature reaches 165°F.
To cook on the grill: preheat grill to medium-high and grill patties until firm and cooked through, about 10 minutes total or until internal temperature reaches 165°F, flipping once.
Assemble burgers: spread the Greek yogurt sauce on top and bottom buns, add the cooked patty, desired greens, and pickled red onions if using, then serve.