Heat 2 tablespoons of olive oil in a large pot or Dutch oven over medium-high heat. Stir in garlic and sauté, stirring occasionally, for 2-3 minutes or just until garlic starts to turn golden brown.

Add chopped onion, carrot, and celery and cook for another 4-5 minutes, stirring occasionally, or until vegetables are soft and tender. We always lightly salt and pepper

Add the diced tomatoes, cannellini beans, 4 cups vegetable broth, bay leaves, rosemary, and thyme to the pan, stir to combine. Add the Parmesan cheese rind, if using, and season with salt and black pepper, to taste. Bring to a boil, then reduce heat to medium-low. Cover and simmer 20-25 minutes.

While soup is simmering, cook pasta according to package directions until al dente. Reserve 1 cup of the cooking liquid before draining pasta. Transfer cooked pasta to a medium bowl and toss with remaining olive oil. Cover and set aside.

Remove cover from pot or Dutch oven and add green beans and the reserved pasta cooking liquid. Stir to combine and continue cooking, uncovered, another 20-25 minutes or until green beans are crisp tender.

Add arugula and additional vegetable broth, if needed, and stir to combine. Season with additional salt and pepper, to taste. Cook for 2-3 minutes or until arugula begins to wilt.