1. In a mixing bowl, combine the salmon, breadcrumbs, egg, mayonnaise, Dijon mustard, lemon juice, basil, salt, and pepper. Mix well.
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, until cooked through.
4. While the patties are grilling, prepare the lemon basil mayo. In a small bowl, combine the mayonnaise, lemon juice, basil, salt, and pepper. Mix well.
5. Serve the salmon burgers on buns with a generous dollop of lemon basil mayo.