In a bowl, combine egg, 1/4 cup cola, crackers, 2 tablespoons salad dressing, Parmesan cheese, and salt. Add beef and mix well. Shape into six 3/4-inch thick patties (mixture will be moist). In a small bowl, combine remaining cola and French dressing; set aside.
Grill patties, uncovered, over medium-hot heat for three minutes on each side. Brush with cola mixture. Grill for eight to 10 minutes longer or until juices run clear, basting with sauce and turning occasionally. Serve on buns.
Note: Diet cola is not recommended for this recipe.