Preheat a grill for high heat. When the grill is hot, lightly oil the grate.
In a medium bowl, mix together the egg, 1/4 cup of cola, crackers, 2 tablespoons of French dressing, and Parmesan cheese. Crumble in the ground beef, and mix well. Form into 6 balls, and pat into 3/4 inch patties. Pour the remaining cola and dressing into a small bowl, and mix well.
Grill burgers for about 3 minutes per side. Brush with the dressing and cola, then grill for 8 to 10 more minutes, basting occasionally.
