Remove skin and bones from fillet; cut fish into 1-inch pieces.
Pulse parsley in a food processor until minced; add green onions, capers, and salmon, and pulse 3 to 4 times or until mixture resembles ground beef. Add egg, 3 tablespoons cracker meal, dillweed, and next 4 ingredients; pulse 3 to 4 times or until blended. Shape mixture into 8 patties. Dredge patties in remaining cracker crumbs, shaking off excess crumbs.
Sauté patties in 2 tablespoons olive oil in a large skillet 3 minutes on each side or until golden brown. Remove from skillet; keep warm.
Whisk together vinegar, mustard, and 1/4 cup olive oil; pour over salad greens and tomatoes, tossing gently. Serve salmon patties over salad with Caper Tartar Sauce.