In a mixing bowl, combine the chopped salmon, breadcrumbs, beaten egg, mayonnaise, Dijon mustard, lemon juice, green onions, basil, salt, and pepper. Mix until well combined.
Form the mixture into 4 patties.
Preheat the grill or skillet over medium-high heat. Cook the patties for about 4-5 minutes on each side, or until cooked through.
While the patties are cooking, prepare the lemon basil mayo by combining the mayonnaise, lemon juice, basil, salt, and pepper in a small bowl. Mix well.
Serve the salmon burgers on buns with a generous dollop of lemon basil mayo.