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 five to seven 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 five 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.