Place crackers in a food processor. Cover and process until coarsely ground. Add half of the salmon, the egg, and 1 tablespoon of the mustard. Cover and process until salmon is ground and mixture is thoroughly combined. Add remaining salmon. Cover and pulse with several on/off turns until salmon is coarsely chopped. With damp hands, shape mixture into four 1/2-inch-thick patties.
Brush patties lightly with olive oil. Grease grill rack. Grill on the rack of an uncovered grill directly over medium heat for 3 minutes per side or until cooked through (160 degrees F). Grill buns, cut side down, the last 1 to 2 minutes of grilling.
Serve salmon burgers on buns with sliced avocado and green onions.