Put half of the salmon in a food processor and pulse until it’s coarsely chopped. Transfer to a medium bowl. Chop the remaining salmon in the same manner, and add to the bowl. Fold in the shallot, capers, mustard, and 3/4 tsp. each salt and pepper. Form into four 3/4-inch-thick patties, set on a large oiled plate and cover, and refrigerate until ready to cook (up to 4 hours).
Meanwhile, make the tartar sauce. In a small bowl, mix the mayonnaise with the relish, dill, lemon juice, and 1/4 tsp. each salt and pepper. Taste and add more lemon juice, salt, and pepper if needed. The sauce will keep for up to one day in the refrigerator.
Heat the oil in a large (12-inch), nonstick skillet over medium-high heat until it’s shimmering. Add the salmon burgers (straight from the refrigerator; the cold temperature will help them hold together), reduce the heat to medium, and cook without touching until they brown at the edges, 2 to 3 minutes. Flip and cook the other sides until they, too, are browned and the burgers are just a little pink in the center (check by slicing into a thicker part of one of the burgers with a paring knife), about 3 to 4 more minutes.
Put the spinach, onion, and 2 tomato slices on each of the buns; top with the burgers and a dollop of the tartar sauce; add the top of the bun and serve.