Spread black olive paste on the rolls and top each burger with leaf lettuce and sliced red onion. Serve the green bean salad alongside.
Place an inch of water in a skillet and bring to a boil.
Place tuna in food processor and coarsely chop, then remove and transfer to a bowl. Add herbs and half of the lemon zest, season with salt and pepper, and combine. Form 4 large patties and reserve on a plate in the refrigerator until ready to cook.
Wash the food processor bowl and return to base. Place olives, capers, anchovies, remaining lemon zest and a drizzle of EVOO in the processor and process into a smooth paste.
To the boiling water, add some salt and the green beans. Cook 5 minutes, then cold-shock the beans under cold, running water. Drain and reserve.
Heat 2 tablespoons EVOO in a nonstick skillet over medium-high heat. Cook tuna burgers 3 minutes on each side for rare, or 6 minutes on each side to cook through completely.
Combine beans with tomatoes and some of the thinly sliced onion. Dress salad liberally with EVOO, lemon juice, salt and pepper. Top with the chopped hard boiled eggs.