How to Make It
Bring a large pot of water to a boil. Fill a bowl with ice water. Using a sharp knife, score a shallow X on the bottom of each tomato. Blanch the tomatoes in the boiling water just until their skins begin to loosen, about 10 seconds. Transfer the tomatoes to the ice water. Peel the tomatoes and halve them lengthwise. Set a strainer over a large bowl. Pry out the tomato pulp and seeds, pressing the pulp through the strainer to extract as much juice as possible. Discard the seeds. Coarsely chop the tomatoes and add them to the bowl with the juices.
Rub the chiles lightly with olive oil and roast them directly over a gas flame or under a broiler, turning frequently, until lightly charred all over. Transfer the chiles to a bowl, cover with plastic wrap and let cool slightly.
Peel and finely chop the chiles. Add them to the tomatoes along with the garlic, the 1/4 cup of olive oil and a generous pinch each of salt and pepper. Let stand for 1 hour.
In a large pot of boiling salted water, cook the linguine until al dente. Drain the pasta well and transfer it to the bowl with the tomatoes. Add the basil and toss well. Serve right away, passing the cheese at the table.