Heat 2 tablespoons olive oil in a large pot over medium heat. Add the onion and pinches of salt and pepper and cook until translucent, about 5 minutes.
Add the garlic, fennel, carrots, and thyme leaves. Stir and cook until the carrots begin to soften, about 10 minutes.
Add the balsamic vinegar, red peppers, beans, tomato paste, broth, and 1/2 teaspoon salt. Simmer until the carrots are tender, 15 to 20 minutes.
Add the simmered soup to a high-speed blender with the remaining 2 tablespoons olive oil and puree until smooth. Season with more salt and pepper, to taste. If you would like more punch, add a few more drops of balsamic, to taste. If you would like a little heat, add 1/2 teaspoon red pepper flakes.
Serve with generous drizzles of olive oil, desired garnishes, and warm baguette.