Make the dressing:
In a food processor, pulse the anchovies, garlic, mustard, and lemon juice until combined. With the machine running, slowly add the olive oil, a few drops at first, and then in a slow, steady stream. Season with salt and pepper
Grill the salad:
Heat the grill. Lightly coat the potatoes, onion, tomatoes, and beans with oil and sprinkle with salt. Lightly brush the tuna with oil and sprinkle with salt and a little pepper. Grill the potatoes over medium-low heat and cook until they’re fork-tender and roasty looking, turning occasionally to keep them from sticking, 30 to 35 minutes. total. Put the onion, tomatoes, and green beans on a medium-high part of the grill. Grill the vegetables until they’re lightly charred and tender, moving the ingredients around so they don’t overcook. The onions will cook in 10 to 15 minutes, the tomatoes in 5 to 8 minutes, and the beans in 4 to 6 minutes. As the vegetables come off the grill, transfer them to a tray or a jellyroll pan, drizzle with dressing, toss lightly, and cover with foil to keep warm.
Meanwhile, grill the tuna over medium-high heat until it’s slightly pink in the center, about 6 minutes per side. Let it cool for a few minutes and slice. Line four salad plates with the lettuce and arrange the grilled vegetables and tuna on top. Garnish with the anchovies, capers, eggs, and olives. Drizzle with a little more dressing, sprinkle with the basil, and serve.