Pasta med kall cheretomatsås

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1. 1
   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.
2. 2
   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.
3. 3
   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.
4. 4
   Divide pasta among shallow bowls and top with more Parmesan and some basil leaves.