Bring a large pot of water to boil. Add the onions and cook for 2 minutes. Drain and cool. Peel the onions and cut off the root ends.
Preheat the oven to 450 degrees F.
Toss the onions, vinegar, 1 tablespoon oil, 1/2 teaspoon of salt and 1/4 teaspoon of pepper in a 13 by 9-inch baking dish. Roast in the oven until the onions are tender and golden, stirring occasionally, about 1 hour.
Meanwhile, whisk the 1/3 cup oil, lemon juice, thyme, and remaining 1/2 teaspoon of salt and 1/4 teaspoon of pepper in another 13 by 9 by 2-inch baking dish. Place the tuna steaks in the marinade and turn to coat. Let stand for 5 minutes on each side.
Oil the grill rack and prepare the barbecue for medium-high heat. Grill the steaks to desired doneness, about 3 minutes per side for medium. Cut the steaks in half. Transfer the steaks to a platter. Spoon the onion mixture around and serve.
Recipe courtesy of Giada De Laurentiis