Cook the pasta: Add the pasta to the boiling water and cook according to the package directions until just shy of tender.

Add the aromatics: Reduce the heat to medium. Add the sliced garlic, Calabrian chili paste, and kosher salt to the skillet. Cook, stirring constantly, until the garlic is fragrant but not browned, about 1 minute.

Reserve water and drain: Just before the pasta is done, scoop out 1 cup of the starchy pasta water from the pot and set it aside. Drain the pasta in a colander.

Build the sauce: Pour 1/2 cup of the reserved pasta water into the skillet with the blistered tomatoes. Bring the liquid to a gentle simmer, stirring to combine the water with the tomato juices and chili paste.

Coat the pasta: Transfer the drained pasta into the skillet. Toss continuously over medium-low heat for 1 to 2 minutes, adding more pasta water a splash at a time if the sauce seems too thick, until the pasta is glossy and well coated.

Add the finishing touches: Remove the skillet from the heat. Stir in the grated Parmesan cheese and torn fresh basil until the cheese melts into the sauce and the basil wilts slightly.

Serve: Divide the pasta among shallow bowls. Top each portion generously with the toasted breadcrumbs and serve immediately.