Heat 3 Tbs. of the oil with the rosemary and red pepper flakes in a 1-quart saucepan over medium heat until shimmering, 1 to 2 minutes. Cool to room temperature.
Heat the remaining 3 Tbs. oil and the garlic in a 6-quart pot over medium heat, turning the garlic occasionally, until golden-brown, 2 to 3 minutes; discard the garlic. Add the peppers and 1/2 tsp. salt and cook, stirring occasionally, until tender and brown in spots, 6 to 8 minutes.
Add the chickpeas and, using the back of a spoon, gently mash some of them against the side of the pot. Stir in 3/4 cup of the cooking liquid, the tomatoes, cheese rind, if using, and 1/2 cup water. Simmer, stirring occasionally, until the flavors have melded, about 20 minutes. (The sauce should be thin but have body. Add the remaining 1/2 cup cooking liquid as needed.)
Meanwhile, bring a large pot of well-salted water to a boil. Add the pasta and cook according to package directions. Reserve 2 cups of the pasta water and drain the pasta.
Add the pasta to the chickpea sauce and cook, tossing gently, until the flavors meld, about 2 minutes. Add pasta water as needed to keep the consistency loose. Season to taste with salt. Discard the cheese rind, if used. Serve drizzled with the rosemary oil and topped with Parmigiano shavings, if you like.