Into a large soup pot over medium-high heat, add the sausage and onion. Cook for 5 minutes, turning frequently until sausages are browned.
Stir in the chicken-style broth, black-eyed peas, potatoes, green beans, Buffalo sauce, ketchup, Worcestershire sauce, garlic, and thyme.
Bring to a boil then reduce heat to low. Simmer for 20 minutes, until potatoes are fork tender.
Whisk together the all-purpose flour and water, making a slurry. Stir into the stew and continue simmering for an additional 10 minutes to thicken slightly. Ladle into soup bowls and serve hot.