Combine egg, bread crumbs, onion, Worcestershire sauce, and pepper in a medium bowl. Add ground beef and mix well.
Using wet hands, shape beef mixture into eight 4-inch round patties. Press one-fourth of the feta cheese into 4 of the beef patties. Top with remaining patties; press to seal edges. (If desired, place patties in a shallow container; cover and chill for 1 hour. Cook within 2 hours.)
Grill patties on the rack of an uncovered grill directly over medium heat for 14 to 16 minutes or until an instant-read thermometer inserted in the burgers registers 160 degrees F, turning once. Serve burgers on top of grilled French bread slices. Top each burger with a little additional crumbled feta cheese, some of the Greek-Style Relish, and remaining grilled French bread slices. Pass remaining relish. Makes 4 servings.
Combine pickled carrots or pickled vegetable, red sweet pepper, pitted kalamata olives, onion, and golden Caesar salad dressing or Italian salad dressing in a bowl. Makes about 3/4 cup.