Preheat the broiler.
Arrange the bell pepper pieces in a single layer on a baking sheet, skin side up. Broil until the skins are black and well-charred, about 8 minutes.
3 large orange bell peppers
Transfer the broiled pepper pieces to a heat-resistant bowl, cover, and let stand until they are cool enough to handle. Remove and discard the skins.
Heat 1½ tablespoons of olive oil in a large pot over medium heat. Once hot, add the onion and vinegar, cover, and cook, stirring occasionally, until the onion has softened, about 8 minutes.
2½ tablespoons extra-virgin olive oil, 1 medium onion, 1½ teaspoons white balsamic
Uncover and sauté until the onions have caramelized, another 8 minutes or so.
Add the garlic and 1 tablespoon of the broth. Cook until fragrant, about 1 minute.
1 clove garlic, 4 cups low-sodium vegetable broth
Add the bell pepper pieces, remaining broth, beans, 1 teaspoon of salt, ¼ teaspoon of black pepper, cayenne pepper, and 1 teaspoon of thyme leaves. Bring everything to a boil, then reduce the heat to medium-low and cover. Simmer until the peppers are very tender, about 25 minutes, stirring occasionally.
14 ounces no-salt-added white beans, 1 teaspoon kosher salt, ¼ teaspoon freshly ground black pepper, ¼ teaspoon cayenne pepper, 1½ teaspoons fresh thyme leaves
While the soup is cooking, preheat the oven to 425°F. Combine the florets, remaining 1 tablespoon of olive oil, and some salt and pepper in a large bowl. Transfer to a baking sheet and roast, stirring once halfway, for about 15 minutes or until the broccoli is tender and browned.
3 cups roughly chopped broccoli florets
Purée the soup in batches in a blender, with a food processor, or using an immersion (stick) blender.
Ladle into bowls and garnish with broccoli and remaining ½ teaspoon thyme leaves.