Slice the onions as thinly as possible. Wash the radicchio and slice it thinly as well. Head the broth. Melt the butter in a pot large enough to prepare the soup in. Add onions and radicchio and sauté over low heat until soft and transparent. Sprinkle in the flour, mixing well. Add one ladleful of broth at a time and mix before adding more. The soup should remain
fairly thick. Allow it to simmer until the vegetables fall apart. Adjust seasonings. Serve over toasted bread, sprinkled with cheese. After removing from heat, you can add a pat of butter or a dash of olive oil.