In a small bowl, combine mayonnaise, tomatoes, parsley, capers, lemon juice and tabasco. Set aside. (This can be prepared in advance and cover it. It will be good up to 6 hours in the fridge).
Put salmon fillet, skin side down, on a cutting board. Beginning by the tight side, with a knife, do a light incision between the skin and the flesh, to be able to hold the skin. Holding the skin firmly and putting the knife down and inclined, pull on the skin from the exterior in a movement of backwards and forwards. Cut the salmons in pieces.
In food processor, mix salmon, 1/4 cup of the reserved tartar mayonnaise, onion, breadcrumbs, salt and pepper until salmon is finely chopped. Shape the mixture in 4 burgers of 3/4 inch thickness. (You can prepare the burgers in advance and put in an hermetic container, separating them with wax paper between each. They will be good up to the next day in the fridge).
Set the barbecue to medium heat. Put the salmon burgers on an oiled grill and close the cover. Cook for about 10 minutes or until golden (flip them at mid-cooking).
Meanwhile, put the slices of bread on the grill of the barbecue and cook 3 minutes or until lightly crispy (flip at mid-cooking). Put 4 slices of bread on a work surface. Garnish each one with a lettuce leaf, then with the salmon burger and remaining of tartar mayonnaise (divided in the 4). Cover with the other slices of bread.