In a mixing bowl, combine the chopped salmon, breadcrumbs, beaten egg, mayonnaise, Dijon mustard, lemon juice, chopped basil, salt, and black pepper. Mix until well combined.
Form the mixture into 4 equal patties.
Preheat the grill to medium-high heat. Grill the salmon patties for about 4-5 minutes on each side, or until fully cooked.
While the patties are cooking, prepare the lemon basil mayo by combining the mayonnaise, lemon juice, chopped basil, salt, and black pepper in a small bowl. Mix well.
Assemble the burgers by spreading the lemon basil mayo on the toasted buns. Place a salmon patty on each bun, and top with lettuce and tomato slices. Serve immediately.