Add half the salmon and the mustard to a food processor and mix it well. Then add the remaining salmon and pulse lightly to combine.
Transfer the salmon mixture to a bowl and add the green onion, breadcrumbs, salt and pepper. Mix with your hands and form into four patties.
Transfer the patties to a plate and cover with plastic wrap. Refrigerate for 2 hours to firm up.
Heat the grill to 400° F. Brush the salmon patties with olive oil on both sides. When hot, add the salmon patties to the grill.
Cook, not touching, for 5 minutes per side. When the burgers reach an internal temperature of 165° F, they are done. You can toast the buns on the grill at the same time.
Remove the burgers from the grill.
Mix the mayonnaise with the garlic in a small bowl.
Serve them on toasted buns with a slice of tomato, lettuce, and a spread of garlic mayo.