Cut off the finocchio tops where they meet the bulb and discard them. Trim away any outer part of the bulb that may be bruised or withered. Slice away ⅛inch or so from the butt end of the bulb. Cut the bulb into very thin vertical slices. Wash the sliced finocchio in several changes of cold water.
Slice off and discard the root or butt end of the radicchio. Pull off and discard any of the outer leaves that are wilted, discolored, or bruised. Cut the radicchio into very thin shreds, and wash in several changes of cold water. Spin or shake thoroughly dry.
Put the sliced onion and the olive oil in a medium skillet or sauté pan, turn the heat on to medium, and cook the onion, turning it over occasionally, until it is completely wilted.
Add the finocchio and radicchio, salt, and liberal grindings of black pepper; cover the pan; and turn the heat down to very low. Cook until the finocchio feels very tender when tested with a fork, about 30 minutes. If you should find, while the vegetables are cooking, that there is not sufficient liquid in the pan to keep them from sticking to the bottom, add 2 or 3tablespoons of water whenever it becomes necessary. When the vegetables are done, however, there should be no runny liquid left in the pan.