Heat olive oil in a large pot over medium heat; cook and stir turkey sausage is browned, about 5 minutes. Add onion and garlic, cooking and stirring until onion is tender, about 5 minutes more. Season with Italian seasoning, salt, and black pepper. Stir in carrots; cook and stir until carrots start to soften, 5 to 7 minutes.
Pour beans, diced tomatoes, beef broth, chicken broth, and tomato sauce into turkey mixture. Stir in bay leaves. Continue to simmer until carrots are tender, about 5 minutes. Add rigatoni pasta; cook and stir until pasta is cooked through but firm to the bite, about 13 minutes.
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