In a food processor pulse the salmon pieces five to ten times until coarsely chopped (but not finely chopped). Transfer the salmon to a large bowl. Add the remaining patty ingredients, including ½ teaspoon salt and ¼ teaspoon pepper, and stir gently to combine. With wet hands, form four patties of equal size, each about ¾ inch thick. Arrange the patties on a platter, cover with plastic wrap, and refrigerate for 1 hour.
In a medium bowl mix the sauce ingredients, including ¼ cup oil, and season with salt and pepper.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Lightly brush the patties on both sides with oil. Grill the patties over direct medium heat, with the lid closed, until nicely browned, 6 to 8 minutes, carefully turning once. During the last minute of grilling time, toast the buns, cut side down, over direct heat.
Build each burger on a toasted bun with lettuce, a patty, tomato slices, and sauce. Serve warm.