Step 1Heat the oven to 300 degrees. Place the tomatoes on a baking sheet and sprinkle half the garlic, plus the thyme, sugar, one-fourth teaspoon salt and one-eighth teaspoon pepper. Bake until the tomatoes are shriveled, 3 to 3 1/2 hours. Cool to room temperature.
Step 2Coarsely chop the tomatoes. In a medium skillet, heat the olive oil over medium heat and cook the remaining garlic until it softens, about 2 minutes. Stir in the tomatoes and the tarragon and cook for 2 minutes. Season with salt and pepper to taste.
Step 1Bring the water, vinegar, sugar, one-half teaspoon salt and one-fourth teaspoon pepper to a boil in a medium saucepan over medium-high heat. Add the onion and cook until slightly softened but still crisp, about 5 minutes. Drain and cool to room temperature. Toss the onion with the olive oil.
Step 1In a medium saucepan, heat the oil and bay leaves to 140 degrees (check with an instant-read thermometer). Pat the fish dry with a paper towel and season with salt and pepper. Slide the fish into the oil and cook, maintaining the temperature at 140 degrees until it is pale gray on the outside but rosy pink in the center, about 5 minutes. Use a slotted spoon to remove the fish.
Step 2To serve, divide the onion mixture evenly among 4 plates. Spoon tomato sauce on top of the onions, then arrange the tuna on top. Season the tuna with salt and serve.