Combine beans, tuna and onion in a large bowl. For dressing, whisk together 3 tablespoons of the mustard, the mayonnaise, 1 tablespoon of the vinegar, 1 teaspoon of the dried dill and the lemon-pepper seasoning in a small bowl. Add dressing to tuna mixture; toss gently to coat. Cover and chill 2 to 4 hours.
Prepare vinaigrette: Combine oil, remaining 1 tablespoon vinegar, remaining 1 teaspoon mustard, salt and the remaining ⅛ teaspoon dried dill in a small screw-top jar. Cover and shake well. Combine spinach and beets in a large bowl. Before serving, pour vinaigrette over spinach mixture; toss gently to coat.
To serve, line four serving plates with spinach mixture. Top with tuna mixture. If desired, sprinkle with fresh dill and/or pepper.
Tip: To cook fresh beets, preheat oven to 400°F. Wash and trim beets; pat dry. Wrap beets in foil and bake 1¼ to 1½ hours or until tender. Cool slightly. Unwrap and slip off skins under cool running water. Place beets in a covered container and chill until needed. Or use packaged refrigerated cooked whole baby beets.