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 oil.
For a charcoal grill, grill patties on the greased rack of an uncovered grill directly over medium coals for 4 to 6 minutes or until fish flakes easily when tested with a fork, turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place patties on greased grill rack over heat. Cover and grill as above.) Add buns to grill, cut sides down, for the last 1 to 2 minutes of grilling to toast.
Spread remaining mustard onto cut sides of buns. Place a salmon burger on bun bottoms. If desired, top with avocado and red onions. Add bun tops.