Poach turkey: Put turkey breast in a 6- to 8-quart pasta pot and add broth and enough water to cover turkey by 1 inch. Remove turkey and bring liquid to a boil. Season with salt and return turkey to pot, then poach at a bare simmer, uncovered, skimming any froth, until an instant-read thermometer inserted 2 inches diagonally into thickest part of breast (do not touch bone) registers 155°F, 1 to 1 1/4 hours. Cool turkey in broth 30 minutes (internal temperature will rise to 170°F).
Make sauce while turkey is poaching: Purée tuna, anchovies, mayonnaise, oil, lemon juice, and water in a blender until very smooth.
Transfer turkey to a cutting board, then bring remaining poaching liquid to a boil.
Add half of beans and cook, uncovered, until crisp-tender, 3 to 5 minutes. Transfer with a slotted spoon to a colander and refresh under cold water. Cook remaining beans and refresh in same way. Add potatoes to broth in pot and simmer until just tender, about 10 minutes. Drain potatoes, reserving broth for another use.
Assemble platter: Thinly slice turkey breast on the diagonal. Arrange beans on a platter and top with potatoes, then drizzle vegetables with some tonnato sauce. Top with turkey slices and more tonnato sauce. Sprinkle dish with capers, olives, and basil. Serve remaining sauce on the side.
Cooks' notes: Turkey may be cooked 1 day ahead. Cool in broth, uncovered, 1 hour, then chill in broth, covered. (Cooling in broth helps keep turkey tender and moist.) If you have a pasta pot with a shallow steamer-basket insert, you can cook turkey in pot, then, after removing it, simmer potatoes in broth while steaming beans in insert. Tonnato sauce can be made 1 day ahead and chilled, covered. Bring to room temperature before serving.