Step 1

Preheat the oven to 300 F. Line a rimmed baking sheet with aluminum foil or parchment paper and coat it with nonstick cooking spray. Place the tomatoes and thyme sprigs on the baking sheet and toss them with 2 tablespoons olive oil. Season with salt and pepper. oast for about 30 minutes, until they are wrinkly and slightly collapsed.

Step 2

While the tomatoes are roasting, bring a large pot of salted water to a boil. While the water is coming to a boil, make the pesto. Place the garlic and basil in a food processor or blender and pulse until everything is roughly chopped. Add 1/3 cup of the oil, a bit of salt and pepper and process, scraping down the sides, part way through, until everything is well blended. If it is very thick, add a bit more olive oil. Add the cheese and pulse until blended in.

Step 3

Cook the cavatappi according to package directions. Reserve 1/2 cup of the cooking water, and drain the pasta. Toss the pasta with the cooking water, remaining 2 tablespoons olive oil (more if it seems dry), and half of the roasted tomatoes (discard the thyme sprigs), and transfer to a serving bowl. Distribute the rest of the tomatoes over the pasta, dollop the pesto on top, and toss. Add more olive oil if desired. Serve hot, warm, or cold.