Prepare an ice bath. Place eggs in a small saucepan, and fill with
enough water to cover by 2 inches. Bring to a boil over high heat; reduce heat, and simmer 7 minutes. Using a slotted spoon, transfer eggs to ice bath to cool. Remove eggs, reserving ice bath. Peel eggs; set aside.
Refill saucepan with water, and bring to a boil over high heat; add a generous amount of salt. Add green beans; cook until tender, about 4 minutes. Drain, and plunge beans into ice bath. Drain; pat dry with paper towels.
Use a sharp knife to slice each tuna steak into two 1 1/2-inch-wide
bars, about 4 inches long. Spread sesame seeds, salt, and pepper on a plate; toss to combine. Dredge tuna in mixture, gently pressing to adhere and turning to coat all sides.
Place a large skillet over medium heat until it feels hot when you
place your palm just above it, about 1 minute. Add 1 1/2 tablespoons olive oil; heat until hot but not smoking, about 1 minute.
Working in batches, carefully place tuna bars in skillet, about 1-inch
apart. Cook until tuna releases easily from the skillet and seeds are fragrant, about 1 minute per side, turning to sear all four long sides. The tuna will be very rare in the center. Increase cooking time for more well-done tuna. Transfer to a plate; let cool. Tuna can be covered in plastic and refrigerated up to 1 day.
Finely chop hard-boiled eggs, and cut beans into 1/2-inch-long
pieces; combine in a bowl. Add olives, and drizzle with the remaining 2 tablespoons olive oil and 2 teaspoons basil oil. Gently toss ingredients together to coat.
Using a sharp knife, slice tuna into 1-inch squares. Drizzle with basil oil. Spoon the egg mixture over each slice; garnish with a small sprig of basil.
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