Bring a medium pot of salted water to a boil. Add the halved baby potatoes and cook for 10 to 12 minutes until tender when pierced with a knife. Remove with a slotted spoon and let cool.

In the same boiling water, cook the green beans for 2 to 3 minutes until bright green and crisp-tender. Transfer immediately to an ice water bath to stop the cooking, then drain and dry thoroughly.

Place the eggs in a saucepan, cover with cold water, and bring to a gentle boil. Lower heat to a steady simmer and cook for 9 minutes. Transfer eggs to ice water for 5 minutes, then peel and quarter them.

Whisk together the olive oil, lemon juice, anchovy paste, minced garlic, salt, and black pepper in a small bowl or shake together in a sealed jar until emulsified.

Line a large platter with butter lettuce leaves. Arrange the cooked potatoes, green beans, quartered eggs, flaked tuna, cherry tomatoes, and olives in separate sections over the lettuce.

Drizzle the vinaigrette evenly over the assembled salad right before serving.