Heat oven to 400 degrees.
Stem, seed and chop the peppers. Chop the zucchini.
Toss the green beans, carrots, peppers and zucchini with 2 tablespoons of olive oil, sugar, 1 teaspoon of salt and ½ teaspoon of pepper.
Line a sheet pan with parchment paper and pour the vegetables in an even layer onto the paper.
Roast the vegetables, stirring a couple of times, for 40 minutes. Remove and set aside.
Chop the onion. Discard the mushroom stems and chop the mushroom caps. Chop the fennel bulb.
Heat a soup pot over medium heat. Add the 2 tablespoons of butter and the chopped onions, mushrooms and fennel.
Cook, stirring occasionally, until the mushrooms are browned and the liquid has evaporated. This will take about 15 minutes. Add the remaining teaspoon of salt and the minced garlic. Cook for about a minute until the garlic is fragrant.
Stir in the tomatoes with their juices, the chicken (or vegetable) stock and the roasted vegetables. Bring the soup to a gentle simmer.
While the soup is simmering, boil a pot of water and cook the pasta just until al dente.
Drain the pasta and add to the soup. Add the beans to the soup and stir well.
Allow the soup to simmer gently for 10 - 15 minutes. Serve the soup topped with chopped fresh basil and grated Parmesan cheese.