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Trim the center stems from the chard leaves, then dice the stems. Set the leaves aside.
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In a soup pot over medium-high heat, warm the olive oil, then add the chard stems, celery, and leek. Cook, stirring often, until the vegetables soften, about 5 minutes. Stir in the garlic and rosemary and cook for 30 seconds. Add the broth and Parmesan rind.
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Cover the pot and bring to a boil. Add the potatoes and rice, reduce the heat, cover, and simmer for 10 minutes.
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Meanwhile, cut the chard leaves crosswise into thin strips. Add the green beans, chard leaves, and cannellini beans to the soup. Bring to a boil, reduce the heat, cover, and simmer until the vegetables are cooked through but haven’t lost their color, about 10 minutes longer. While the soup is cooking, prepare the optional pistou.
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When the soup is done, remove the lid and let the soup stand off the heat for 10 minutes to settle the flavors. To serve, season with salt to taste, ladle into bowls, and swirl a spoonful of pistou into each portion.