Boil the water: Fill a large pot with water and add enough salt to make the water taste lightly seasoned. Set the pot over high heat and bring the water to a full boil.

Cook the pasta: Add the pasta to the boiling water in the pot and stir for a few seconds so it does not stick. Cook it according to the package directions until it is tender with a little firmness in the center.

Reserve the pasta water: Before you drain the pasta, scoop 1 cup of the pasta water into a mug or heat-safe measuring cup so you can use it to loosen the sauce later.

Reserve water and drain: Right before draining the pasta, carefully scoop out about 1 cup of the starchy pasta cooking water and set it aside in a heatproof mug or bowl. Drain the pasta in a colander.

Cook the garlic: Heat the extra-virgin olive oil in a large skillet over medium heat. Add the thinly sliced garlic and cook for about 1 minute, stirring constantly, until fragrant but not browned.

Simmer the liquids: Stir the balsamic vinegar and 1/2 cup of the reserved pasta water into the skillet with the burst tomatoes. Let the liquid bubble and simmer for 2 minutes to form a cohesive, slightly thickened sauce.

Toss the pasta: Add the drained linguine to the skillet. Toss the pasta continuously for 1 to 2 minutes, adding more reserved pasta water a splash at a time if the sauce seems too dry, until the noodles are evenly coated.

Finish and serve: Remove the skillet from the heat. Stir in the torn fresh basil, freshly cracked black pepper, and grated Parmesan cheese, then serve immediately.