Step 1In a blender, puree the basil with one-half cup of the oil, the garlic and lemon juice until completely smooth. Scrape into a bowl and let the mixture stand while continuing with the next steps. Stir well before using.
Step 2In a large skillet, heat 1 tablespoon of the remaining oil over medium-high heat. Add the tomatoes and cook, stirring gently, until soft, about 6 to 7 minutes. Stir in the sugar, 2 tablespoons of the vinegar, and salt to taste and cook for another minute. With a slotted spoon, transfer the tomatoes to a sieve set over a bowl and let the liquid drain off for a few minutes. Cool the tomatoes, then refrigerate for about 1 hour.
Step 3Heat the oven to 425 degrees. Line a cookie sheet with foil and brush the foil lightly with oil. Press the peppers gently to flatten them and place them skin-side-up on the cookie sheet. Brush with a little oil and roast until the peppers are tender and lightly charred, about 35 minutes.
Step 4Transfer the peppers to a large bowl, cover with plastic wrap and let stand for 15 minutes. Peel the peppers and cut into thin strips. Place them back in the bowl and toss with 1 tablespoon of the olive oil, the remaining 1 tablespoon of the vinegar and a little salt. Let stand for about 30 minutes.
Step 5Line a 9-inch pie plate with plastic wrap so that you have some overhang on two sides. Arrange half of the roasted peppers on the bottom, scatter half of the tuna on top, drizzle with about 2 teaspoons basil puree and top with half of the tomatoes. Make a second layer using the remaining peppers, tuna and tomatoes, drizzling again lightly with the basil puree.
Step 6Place a large serving plate over the pie plate, invert and remove the plastic.
Step 7In a small bowl, whisk the olive paste with the remaining 3 tablespoons of the oil. Drizzle the sides of the serving plate decoratively with the basil puree (you might not need all of it) and dot with olive puree, then drizzle a little of each puree over the top of the salad.