1 can (15.5 ounces) chickpeas, drained and rinsed
1/2 cup Kalamata olives, pitted
2 tablespoons tomato paste
3 cloves garlic, thinly sliced
1 6-inch sprig rosemary
3 tablespoons extra-virgin olive oil, plus more for serving
1/4 teaspoon red-pepper flakes, plus more for serving
Coarse salt and freshly ground pepper
1/2 cup grated Parmigiano-Reggiano (about 1 1/2 ounces), plus more for serving
2 cups baby arugula (about 2 ounces)
Combine pasta, chickpeas, olives, tomato paste, garlic, rosemary, oil, pepper flakes, and 5 cups water in a large straight-sided skillet. Season generously with salt and pepper. Bring to a boil, then cook over medium-high heat, stirring occasionally, until pasta is al dente and liquid is reduced to a sauce that coats pasta, 12 to 15 minutes. Remove skillet from heat, discard rosemary, and stir in cheese. Add more water, a few tablespoons at a time, to thin sauce if necessary. Divide pasta and arugula among bowls and serve immediately, drizzled with oil and sprinkled with cheese and pepper flakes.