Combine 7 garlic cloves, finely grated, 1½ lb. cherry tomatoes (about 2 pints), halved, 1½ cups (lightly packed) coarsely chopped basil, ½ cup extra-virgin olive oil, 2 Tbsp. Calabrian chile paste, 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and ¾ tsp. freshly ground pepper in a large bowl. Using your hands or a potato masher, crush tomatoes to release their juices and break up into small pieces. Let cherry tomato mixture sit at room temperature at least 10 minutes and up to 1 hour.

Meanwhile, cook 12 oz. dried lumache (snail shells) or other medium shell pasta in a large pot of boiling heavily salted water, stirring occasionally, until al dente. Drain, reserving 1 cup pasta cooking liquid. Shake pasta to remove excess water; return to pot.

Immediately add 3 oz. Parmesan, finely grated (about 1½ cups), and 5 Tbsp. unsalted butter, cut into ½" pieces, to pasta and vigorously stir until butter and cheese are melted. (If pasta seems dry, add some cherry tomato mixture a spoonful at a time as needed and keep stirring until cheese is melted and emulsified.) Add cherry tomato mixture and toss, adding reserved pasta cooking liquid a splash or so at a time as needed, until sauce is creamy and coats pasta. Taste pasta and season with more salt and pepper if needed.

Divide pasta among shallow bowls and top with more Parmesan and some basil leaves.