Brush grill with oil; prepare barbecue (medium heat). Mix mayonnaise, 1 cup chopped basil, cornichons, brine, and onions in small bowl. Season basil mayonnaise with salt and pepper.
Mix chicken and 1/2 cup basil mayonnaise in large bowl. Using wet hands, shape chicken mixture into four 1/2-inch-thick patties. Sprinkle with salt and pepper. Brush bread with oil.
Grill bread until crisp and golden, about 1 minute per side. Transfer bread to plates. Grill burgers until firm to touch and cooked through, about 5 minutes per side.
Spread bread slices thickly with basil mayonnaise. Top each bread slice with tomato, 2 basil leaves, burger, and dollop of basil mayonnaise. Garnish with remaining chopped basil.