Coarsely chop the tomatoes, making sure to reserve the juice. Place tomatoes and juice in large bowl. Add the vinegar, 3 tablespoons of the extra-virgin olive oil, and the basil. Add the tablespoon of olive oil to a saute pan over medium high heat. Saute the garlic for 3 minutes, stirring. (Do not allow it to brown). Add to tomato mixture. Season to taste with salt and pepper. Let mixture sit at room temperature for at least 4 hours. When ready to serve, cook linguine in a large pot of boiling, salted water. When pasta is al dente, drain in a large colander, then transfer to a large bowl. Toss linguine with a tablespoon of extra-virgin olive oil and then with the Parmigiano-Reggiano. Divide linguine among 4 to 6 bowls. Top each bowl with about a cup of tomato chunks, retrieved from the tomato mixture with a slotted spoon. Ladle the remaining liquid around.