In food processor container, place salmon, egg, breadcrumbs, garlic, soy sauce, basil, and lime zest, and pulse several times until you have a chunky mixture. Form into 4 patties.
In large bowl, whisk together lime juice, fish sauce, sesame oil, and sugar. Stir in tomatoes, mango, green onions, chili pepper, and cilantro.
In large skillet over medium-high, heat oil. Place salmon burgers in skillet and cook for 3 minutes on each side, or until crispy on the outside but just barely cooked through in middle, to an internal temperature of 145 F. Alternatively, you can prepare the burgers on an outdoor grill over medium heat, making sure grill grates are well greased.