Heat oil in a dutch oven (or large pot). Add sausages and break them up with a wooden spoon.
Cook until sausages have browned, about 5-7 minutes. Add onions and garlic and continue to cook until onions become translucent. Sprinkle cornmeal over sausage onion mixture and mix to combine. Continue to cook for an additional 5 minutes, stirring frequently. Add the broth and Swiss chard, and bring to a gentle boil, then reduce to a simmer and cook partially covered for 35-40 minutes.
To serve, ladle soup into bowls and top with a poached egg and finish with a drizzle of olive oil.