Bring a medium saucepan of salted water to a boil. Meanwhile, heat 2 tablespoons olive oil in a large Dutch oven or wide pot over medium-high heat. Add the cauliflower in a single layer and cook, without stirring, until browned, about 5 minutes. Stir well, season with salt and cook, stirring occasionally, until browned all over and tender, 7 to 10 minutes. Push to one side of the pot and add the remaining 1 tablespoon olive oil and the chickpeas. Cook the chickpeas, stirring, until toasted, about 5 minutes. Stir in the garlic, red pepper flakes and a pinch of salt.
Meanwhile, add the cavatelli to the boiling water and cook as the label directs. Reserve 1 cup cooking water and then drain. Add the pasta, butter and 1/2 cup cooking water to the Dutch oven and cook, stirring, until the pasta and cauliflower are well combined. Add the baby greens in two batches, stirring, and cook until just wilted, 30 seconds to 1 minute. Remove from the heat.
Stir 2 tablespoons cheese into the pasta, adding more cooking water if the pasta is too dry; season with salt. Divide among bowls and top with the remaining 2 tablespoons cheese.