Combine 3 tablespoons of the lemon juice, 4 teaspoons of the garlic, the olive oil, the oregano, ½ teaspoon of the salt, and ½ teaspoon of the pepper in a zip-top bag. Add the chicken; seal the bag and shake to coat the meat. Refrigerate it for at least 1 hour or up to 4 hours.
Coat a grill pan or large heavy skillet with cooking spray and heat it to medium-high. Remove the chicken from the bag and discard the marinade. Add the cutlets to the pan and cook until no longer pink in the center (165°F internal temperature), 3 to 4 minutes per side.
Meanwhile, combine the yogurt, cucumber, dill, and remaining 1 tablespoon of lemon juice, 1 teaspoon of garlic, ¼ teaspoon of salt, and ¼ teaspoon of pepper in a small bowl. Spread the mixture on one side of all the slices of bread.
Layer the cheese, cooked chicken, tomato, lettuce, and onion on four of the bread slices. Top with the remaining bread slices, compressing the sandwiches.
Cook the sandwiches in the pan under a heavy weight (such as another heavy skillet) until they are browned, 3 to 4 minutes per side. Cut each sandwich in half on the diagonal.
Tip: If you don’t have fresh or dried oregano, use fresh parsley or dried thyme.