Squeeze juice and seeds from tomato halves into a fine-mesh sieve over a bowl, reserving juices; discard seeds. Finely chop tomatoes.
Heat 2 tablespoons oil in a large nonstick skillet over medium heat. Add garlic to pan; cook 30 seconds or just until garlic begins to brown, stirring constantly. Add tomatoes, reserved juices, 1/2 teaspoon salt, and pepper. Increase heat to medium-high; cook 15 minutes or until liquid almost evaporates, stirring occasionally.
Bring 6 quarts water and remaining 2 tablespoons salt to a boil in an 8-quart pot. Add pasta to pot; stir. Cover; return water to a boil. Uncover and cook 8 minutes or until pasta is almost al dente. Drain pasta in a colander over a bowl, reserving 1/2 cup cooking water.
Add hot pasta and reserved cooking water to tomato mixture. Cook 5 minutes or until sauce is thick and pasta is al dente, tossing to combine. Remove from heat. Sprinkle with basil; toss. Place 1 cup pasta mixture on each of 6 plates. Drizzle each serving with 1/2 teaspoon of remaining oil; sprinkle each with 1 tablespoon cheese.
Fantastic pasta you can buy: Imported Italian artisanal dried pasta can be hard to find, but it's worth the search when you want authenticity. It's made with the highest quality wheat, which gives it superior flavor, and the dough is extruded through bronze dies, resulting in beautiful shapes with porous surfaces to which sauce readily clings. It's also slowly dried, producing a sturdier product.
Look at local specialty markets and online grocery stores for quality imported artisanal pasta. A few of the top brands:
Latini: Look for their heirloom wheat Senatore cappelli or Taganrog riccioli (curls) pastas.
Benedetto Cavalieri and Martelli: Both produce excellent spaghetti, bucatini, and short pastas like penne and fusilli.
Pasta di Gragnano, Setaro, and Gerardo di Nola: All make traditional shapes from the Neapolitan region of Campania, like candele (candle-shaped, long hollow tubes, meant to be broken), calamari (short, wide rings), paccheri (oversized rigatoni), and fusilloni (large corkscrews).