Bring a large pot of salted water to a boil. Adjust an oven rack to the upper-middle position and heat the broiler to high. Line a large rimmed baking sheet with foil and rub it with oil. Toss the tomatoes with 1 tsp. oil and about 1/4 tsp. salt. Spread them on the baking sheet. Broil, shaking the pan occasionally, until they're cracked, very soft, and shrunken, about 20 minutes; they'll be blackened in places.
Meanwhile, chop the garlic and put it in a small saucepan, along with 3 Tbs. oil and the red pepper flakes (see "An easy way to peel a lot of garlic") . Bring to a simmer over medium heat and cook until the garlic just begins to turn golden, about 1 minute. Remove from the heat and let the oil steep.
Stack the basil leaves on top of one another and roll them into a cigar shape. Slice across the cigar to create thin, but not ultra-thin, ribbons (about 1/4 cup).
When the tomatoes are done, cook the pasta until tender, 3 to 4 minutes. Drain the pasta and return it to the pot. Immediately toss it with the garlic oil (and the garlic) and 1/2 to 1 tsp. salt. Grind lots of fresh pepper over the pasta, add the broiled tomatoes and the basil, and toss to combine well. Add half the Parmigiano, toss again, and immediately turn out into warm bowls. Top with the remaining cheese.