In a medium bowl mix the mayonnaise ingredients until well blended.
Prepare the grill for direct cooking over medium heat (180° to 230°C).
Prepare the grill for cooking over medium heat (200°C).
Grate the onion on the large holes of the box grater. Squeeze the excess moisture from the grated onion with paper towels. Put the onion in large bowl with the remaining patty ingredients and gently combine. Arrange six squares of foil on a work surface. Drop equal amounts of the mixture onto each square. Using wet fingers, press firmly into patties approximately the size and shape of the bread slices and about 12 millimeters thick. Lightly brush each patty with oil.
In a small bowl mix the ketchup, tomato puree, and Worcestershire sauce. Set aside.
Brush the cooking grates clean. Lightly brush the bread slices on both sides with oil and grill them over direct medium heat, with the lid closed, until lightly toasted, about 1 minute, turning once. Remove from the grill. Place the patties, foil side up, on the cooking grates and carefully peel off the foil. Grill over direct medium heat, with the lid closed, for about 5 minutes. Turn the patties over and spread each one with about 1 tablespoon of the ketchup mixture. Continue grilling, with the lid closed, until the meat is firm and fully cooked but still moist, 4 to 5 minutes more. Remove from the grill.
Brush the cooking grates clean. Lightly brush the bread slices on both sides with oil and grill them over medium heat, with the lid closed, until lightly toasted, about 1 minute, turning once. Remove from the grill. Place the patties, foil side up, on the cooking grates and carefully peel off the foil. Grill over medium heat, with the lid closed, for about 5 minutes. Turn the patties over and spread each one with about 1 tablespoon of the ketchup mixture. Continue grilling, with the lid closed, until the meat is firm and fully cooked but still moist, 4 to 5 minutes more. Remove from the grill.
Spread some of the mayonnaise on six slices of the bread. Top each with a meat loaf patty and the remaining bread slices. Serve with the remaining mayonnaise.
