Warm oil in a large pot over medium heat. Once the oil is shimmering, add the chopped onion, carrot. Cook, stirring often, until the vegetables have softened and the onions are turning translucent, about 7 to 10 minutes.
Add the seasonal vegetables, garlic. Cook until fragrant while stirring frequently, about 2 minutes.
Pour in the diced tomatoes and their juices, broth and water. Add the salt, bay leaves. Season generously with freshly ground black pepper
Raise heat to medium-high and bring the mixture to a boil, then partially cover the pot with the lid, leaving about a 1” gap for steam to escape. Reduce heat as necessary to maintain a gentle simmer.
Cook for 15 minutes, then remove the lid and add the pasta, beans and greens. Continue simmering, uncovered, for 20 minutes or until the pasta is cooked al dente and the greens are tender.
Remove the pot from the heat, then remove the bay leaves. Taste and season with more salt and pepper