Thaw salmon, if frozen. Rinse salmon; pat dry with paper towels. Cut into 1-inch pieces. Place salmon, half at a time, in a food processor. Cover and process until finely chopped. In a large bowl combine egg, green onions, salt, and black pepper. Add chopped salmon; mix gently until combined. Using damp hands, shape mixture into six 1/2-inch-thick patties. Cover and chill for at least 30 minutes.
Meanwhile, in a small bowl combine mayonnaise and 2 to 3 tablespoons Asian chili sauce. Cover and chill until needed.
For slaw, in a large bowl whisk together 1 tablespoon vegetable oil, vinegar, soy sauce, sugar, and sesame oil. Add shredded cabbage and cilantro; toss to coat.
Lightly brush both sides of salmon patties with additional vegetable oil.
For a charcoal or gas grill, place patties on the rack of a covered grill directly over medium heat. Grill for 6 to 8 minutes or until done (160 degrees F), turning and brushing once with sweet chile sauce halfway through grilling.
To serve, spread cut sides of buns with mayonnaise mixture. Fill with salmon burgers, slaw, and cucumber slices.