Preheat oven to 400°. On a large rimmed baking sheet, toss potatoes and beans with 2 teaspoons olive oil; season with half of the salt and pepper. Roast 15 to 20 minutes or until beans are browned and tender. Transfer beans to a plate, toss potatoes again, and continue roasting 10 minutes or until tender. Add potatoes to plate with beans.
In a small bowl, whisk together lemon juice, remaining 2 tablespoons olive oil, capers, parsley, and remaining salt and pepper. Divide lettuce among 4 plates; top each with 1/4 of the tuna, olives, tomatoes, potatoes, and beans.
Spray a large nonstick skillet with cooking spray. Crack two eggs into skillet. Cook until whites are cooked and yolks are just set (about 2 minutes). Repeat with remaining eggs. To serve, drizzle each salad with dressing and place one egg on top.