1. In a mixing bowl, combine the chopped salmon, breadcrumbs, beaten egg, mayonnaise, Dijon mustard, lemon juice, chopped basil, salt, and pepper.
2. Form the mixture into 4 patties.
3. Preheat the grill to medium-high heat. Grill the patties for about 4-5 minutes on each side, or until cooked through.
4. In a small bowl, mix together the mayonnaise, lemon juice, chopped basil, salt, and pepper to make the lemon basil mayo.
5. Serve the salmon burgers on buns with a dollop of lemon basil mayo.