Bring a large pot of salted water to a boil over high heat. Cook pasta according to package directions. Drain pasta and return to cooking pot, reserving 1 cup of cooking liquid.
While pasta cook, in a large skillet, heat one tablespoon olive oil over medium-high heat until shimmering. Add half the garlic and cook, stirring frequently with a wooden spoon, until the garlic is soft and lightly brown, about 2 minutes. Add half the spinach and cook until tender, about 1 minute. Transfer the cooked spinach to a bowl. Repeat with another tablespoon of olive oil, remaining garlic, and spinach
Add the spinach, tomatoes, olives, and remaining olive oil to the pasta and stir to combine. If the sauce seems too thick, loosen it by adding pasta water a little at a time until desired consistency is reached. Add three-quarters of the grated ricotta salata. Stir to combine.
Season to taste with kosher salt and freshly ground black pepper. Pour pasta into a large serving bowl or spoon onto individual plates. Top the pasta with the remaining ricotta salata and basil.