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Cook the sausage and cut them into small pieces or slice very thinly. Set aside.
Melt the butter in a 4-quart pan and add the chopped leeks. Sauté over medium heat until soft, then add the chicken broth and the beef broth. Cover and bring to a boil.
Stir in the cabbage, basil, tomatoes, and garlic. Cover, return to a boil, lower the heat, and uncover. Simmer uncovered for 3 to 5 minutes and add the sausage.
Simmer for 2 to 3 minutes longer, uncovered, until the cabbage is crisp-tendered. Add salt and pepper and serve immediately.