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Heat the oil in a large stockpot or Dutch oven over medium heat. Add the chicken sausage and cook, breaking up large pieces with a spoon, until the sausage is no longer pink and is just beginning to brown in spots. Transfer to a plate.
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Add the onion, carrots, celery, and mushrooms to the pot and cook until the vegetables have softened and the onion is translucent, about 5 minutes. Add the garlic and cook for one minute more. Season with 1 teaspoon salt and 1/2 teaspoon pepper. Add the sausage and any juices back to the pot. Pour in the chicken stock, diced tomatoes, and white beans. Add the Parmesan rind (if using). Bring the soup to a boil, then reduce the heat to a simmer and partially cover. Simmer the soup for 20 minutes.
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Remove the Parmesan rind and discard. Add the escarole to the soup, cover, and simmer for a minute or two, just until wilted. Taste the soup and season with more salt and pepper as needed. Ladle into bowls and sprinkle with Parmesan cheese before serving.