Stir together tomatoes, sugar, and salt in small bowl. Bring 6 cups water to boil in a medium saucepan over medium-high heat. Add potatoes and 3 tablespoons salt and cook until potatoes are easily pierced with paring knife, 12 to 15 minutes. Using tongs, transfer potatoes to a cutting board, leaving water in saucepan.
Return water to boil. Add green beans and cook until just tender, 4 to 5 minutes. While green beans cook, fill a medium bowl halfway with ice and water. Using tongs, transfer green beans to ice water, leaving water in the saucepan. Let beans cool, about 5 minutes. While green beans cool, gently flatten each potato with side of chef's knife to ½- to ¾-inch thickness (it's OK if some potatoes crumble).
Heat oil in 10-inch nonstick skillet over medium-high heat until shimmering. Add potatoes and cook until brown and lightly crisped on both sides, 2 to 3 minutes per side. Meanwhile, return water to boil over medium-high heat. Add eggs, cover, and cook for 8 minutes (eggs needn't be submerged).
Line a cutting board with double layer of paper towels. Transfer crisped potatoes to half of paper towel and season with salt to taste. Transfer green beans to remaining half of paper towel to drain, leaving cold water in bowl.
Divide tomatoes, tuna, and beans between 2 plates, piling them in separate mounds. Transfer cooked eggs to cold water to chill for 30 seconds. Peel and halve eggs, and place 3 halves on each plate. Divide potatoes between plates. Drizzle each salad with 2 tablespoons vinaigrette. Sprinkle salads with olives and capers and serve, passing extra vinaigrette.