Heat 2 tablespoons of the oil in a medium skillet over medium heat. Add the celery and onions and cook until softened, 3 to 4 minutes. Set aside.
Combine the salmon in a food processor along with half of the breadcrumbs, the onion and celery mixture, the green onions, mustard, salt, pepper and egg. Pulse until well combined, 15 to 20 times. Heat the remaining oil in a large skillet over medium heat. Form the salmon mixture into 8 small patties (about 2 tablespoons each) and dust with the remaining breadcrumbs. Cook until browned, 4 minutes per side.