In a medium bowl, mix together eggs, ground beef, 1/2 cup cola, crackers, garlic powder, onion powder, salt, pepper, 1/4 cup French salad dressing, and Parmesan cheese. Mix gently. Cover with plastic wrap. Refrigerate at least 1 hour. Pour remaining cola and salad dressing into a separate bowl, and mix well (refrigerate until ready to grill.) When ready to grill, form meat into 12 patties.
Oil grill grate. Heat grill to medium. Grill burgers 5-6 minutes per side. Baste with salad dressing mixture every 2 minutes. In the last couple of minutes, top with cheese slices. Serve with sliced tomatoes and lettuce.