In a large pot, heat the oil over medium-high. Add the onion and carrots and cook, stirring occasionally, until the vegetables soften slightly, 4 to 5 minutes. Stir in the Swiss chard and cook, stirring often, until the chard wilts, about 2 minutes. Add the tomatoes and cook, stirring constantly, until slightly reduced, about 1 minute. Pour in the broth and bring to a boil. Reduce the heat to medium and simmer the soup, stirring often, until the flavors blend, about 10 minutes. Season with salt and pepper.
Stir in the pasta, increase the heat to medium-high and bring to a boil. Cover, reduce the heat to medium-low and cook, stirring occasionally, until the pasta is tender, about 12 minutes. Stir in the beans and cook until heated through, about 2 minutes. Season.
Preheat the broiler. Cut each bread slice in half crosswise and top with the cheese. Place on a foil-lined baking sheet and broil until the cheese melts, about 1 minute. Serve the cheese toast with the soup.