Whisk together vinegar, shallot, mustard, garlic paste, and anchovy paste in a small bowl until combined well, then add oil in a slow stream, whisking until emulsified. Whisk in thyme, basil, and salt and pepper to taste.

Cook beans in a 4- to 6-quart pot of boiling salted water , uncovered, until crisp-tender, 3 to 4 minutes, then immediately transfer with a slotted spoon to a bowl of ice and cold water to stop cooking. Add potatoes to boiling water and simmer, uncovered, until tender, 15 to 20 minutes, then drain in a colander. Halve potatoes while still warm (peel if desired) and toss with 2 tablespoons dressing in a bowl, then cool.

Prepare grill for cooking. If using a charcoal grill, open vents on bottom of grill, then light charcoal. Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds. If using a gas grill, preheat burners on high, covered, 10 minutes, then reduce heat to moderately high.

Brush tuna with oil and season with salt and pepper, then grill on lightly oiled rack, uncovered, turning over once, until browned on outside but still pink in center, 6 to 8 minutes total. Let tuna stand 3 minutes, then break into large (3-inch) pieces. Transfer tuna to a large platter and drizzle with 2 to 3 tablespoons dressing and top with capers.

Transfer potatoes to platter with tuna, reserving bowl. Drain beans and pat dry. Toss beans in bowl with 1 tablespoon dressing and salt and pepper to taste, then transfer to platter. Toss lettuce in bowl with 2 tablespoons dressing and salt and pepper to taste, then transfer to platter. Toss tomatoes in bowl with 1 tablespoon dressing and salt and pepper to taste, then transfer to platter.

Arrange olives and eggs on platter and sprinkle salad with parsley and/or basil. Serve salad with remaining dressing on the side.