To make vinaigrette: Whisk oil, vinegar, shallot and garlic in a small bowl. Season with salt and pepper. Set aside.
To make salad: Rub oil all over tuna. Sprinkle the tuna with salt and pepper. Press about 1 teaspoon thyme leaves into each side.
Heat a large heavy skillet (not nonstick) over high heat until very hot. Add the tuna and sear until browned and crusted on the bottom, about 2 minutes. Turn the tuna over and cook until browned, about 2 minutes more for medium-rare.
Meanwhile, combine arugula, tomatoes, basil and mint in a large bowl. Toss with reserved dressing. Taste and adjust seasoning.
Slice the tuna into 1/4-inch-thick slices. Divide the salad among 4 plates, mounding it to one side. Fan the tuna slices around the salad. Serve immediately.
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