3 medium onions, chopped
1 red bell pepper, chopped
1/2 cup extra-virgin olive oil plus more
freshly ground pepper
4 garlic cloves, 3 finely grated, 1 halved
2 teaspoons tomato paste
4-6 1"-thick slices grilled or toasted ciabatta
8-10 tablespoons finely grated Parmesan, divided
2 15-ounce cans cannellini (white kidney) beans, rinsed and drained
4 cups vegetable broth, divided
1 cup cherry tomatoes, halved
2 tablespoons chopped flat-leaf parsley
Pulse onions in a food processor until finely chopped but not puréed (you should have about 2 cups). Transfer to a medium bowl. Pulse bell pepper in processor until finely chopped but not puréed (you should have about 1 cup); add to bowl and mix well.
Heat oil in a large heavy skillet over medium heat. Add onion mixture (it may splatter) and season with salt and pepper. Simmer, stirring often, until vegetables are completely softened, about 30 minutes. Add finely grated garlic and tomato paste and cook, stirring often, until tomato paste begins to turn deep red, about 3 minutes. Measure 1/2 cup soffritto and set aside; reserve skillet. Transfer remaining soffritto to a container and let cool completely, uncovered. Cover and store in refrigerator for up to 4 days or freeze for up to 3 months.
Meanwhile, preheat oven to 375°F. Rub bread slices with cut sides of remaining garlic clove. Place bread on a baking sheet and sprinkle 1 tablespoon Parmesan over each slice. Toast until cheese begins to brown, about 5 minutes. Set aside.
Heat reserved 1/2 cup soffritto and beans in same skillet over medium-high heat. Cook, stirring often, until heated through, about 1 minute. Stir in 3 cups broth; bring to a boil. Simmer, scraping up browned bits, until liquid is slightly thickened, 3-4 minutes. Add tomatoes and remaining 1 cup broth; simmer until tomatoes are tender, 3-4 minutes. Stir in 2 tablespoons Parmesan. Season with salt and pepper.
Divide bread among bowls. Top with some bean mixture and broth. Garnish with remaining 2 tablespoons Parmesan and parsley. Drizzle with oil, if desired.