Preheat broiler. Arrange bell pepper pieces in a single layer on a baking sheet, skin side up. Broil until skins are black and well-charred, about 8 minutes. Transfer pepper pieces to a heat-resistant bowl and cover. Let stand until cool enough to handle. Remove the skins and discard.
Heat 1 1/2 tablespoons olive oil in a large pot over medium heat. Add onion and vinegar, cover, and cook, stirring occasionally, until the onion has softened, about 8 minutes. Uncover and sauté until the onions have caramelized, another 8 minutes or so.
Add garlic and about 1 tablespoon of the broth. Cook until fragrant, about 1 minute. Add bell pepper pieces, remaining broth, beans, 1 teaspoon salt, 1/4 teaspoon black pepper, cayenne pepper, and 1 teaspoon of thyme leaves. Bring to a boil, then reduce heat to medium-low, cover, and simmer until peppers are very tender, about 25 minutes. Stir occasionally.
While the soup is cooking, roast the broccoli. Preheat oven to 425°F. Combine florets, remaining 1 tablespoon olive oil, and some salt and pepper in a large bowl. Pour onto baking sheet. Roast, stirring once halfway, for about 15 minutes or until broccoli is tender and browned.
Purée the soup in batches in a blender or food processor (follow manufacturer's directions for hot ingredients) or using an immersion (stick) blender.
Ladle into bowls and garnish with broccoli and remaining 1/2 teaspoon thyme leaves.