In a mixing bowl, combine the chopped salmon, breadcrumbs, beaten egg, mayonnaise, Dijon mustard, lemon juice, lemon zest, chopped basil, salt, and pepper. Mix well.
Form the mixture into 4 patties.
Heat a grill or skillet over medium-high heat. Cook the patties for about 4-5 minutes on each side, or until fully cooked through.
In a small bowl, whisk together the mayonnaise, lemon juice, lemon zest, chopped basil, salt, and pepper to make the lemon basil mayo.
Serve the salmon burgers on buns with a generous dollop of lemon basil mayo.