Place the tomatoes on a large rimmed baking sheet. Drizzle with 2 tablespoons olive oil and add the sliced garlic, ½ teaspoon of kosher salt, and black pepper. Toss to coat. Roast for 20 to 25 minutes, stirring once, until the tomatoes are softened, blistered, and beginning to burst. Set aside.
While the tomatoes roast, bring a large pot of generously salted water to a boil. Cook the fusilli according to the package directions until just al dente.
Before draining, reserve 1 cup pasta water. Drain the pasta. Immediately reserve 1 cup of the cooked pasta for making the crispy pasta topping.
Return the remaining pasta to the pot. Add 2 tablespoons olive oil, ½ teaspoon each of salt, pepper, and the Parmesan. Add ¼ cup of the reserved pasta water and toss until the Parmesan melts into a light, glossy sauce. Add more pasta water, a tablespoon at a time, if needed.
While the pasta is cooking, heat the remining 2 tablespoons olive oil in a large skillet over medium-high heat. Add the reserved 1 cup cooked pasta and spread it into a relatively even layer. Sprinkle with a large pinch of salt. Cook without stirring for 3 to 4 minutes, or until the bottoms are golden and crisp. Stir and continue cooking for another 3 to 5 minutes, stirring occasionally, until the pasta is deeply golden and crispy in spots. Transfer to a plate lined with paper towels.
Add the roasted tomatoes and their juices to the cooked pasta and gently toss to combine. Transfer the pasta to a large serving platter or individual bowls. Add the fresh mozzarella pearls and spoon the chili crisp over the top. Scatter the crispy pasta over everything, followed by the fresh basil. Finish with more freshly grated Parmesan and a little flaky sea salt, if desired.