Butter and toast the slices of French bread under the broiler. Set aside. Turn broiler OFF, and set your soup bowls in the oven to pre-warm them, while you prepare the soup.
In a large, heavy saucepan, cook onions and garlic in oil until onion is tender. Don't let the garlic burn! Add mushrooms and cook another 8 - 10 minutes, stirring frequently. Add broth and tomato paste. Bring to a light boil. Remove from heat.
In a small bowl, combine egg yolks, 1/3 cup Parmesan cheese and parsley, stirring until well-mixed.
Gradually and slowly add 1 cup of the hot liquid to the egg yolk mix, stirring constantly with a whisk. Then, after the one cup of hot liquid is well-incorporated into the egg yolks, return the egg yolk mixture to the hot soup base mixture, stirring constantly with a whisk. You are doing this so you'll introduce the hot mixture slowly into the egg yolks, and they won't cook instantly and make lumps of hard-boiled yolk. Be sure to stir constantly.
Stir well, and return it all to the heat. Bring it all up to temp., until it JUST reaches a soft boil. Add the wine, stir just until heated through, and remove from heat.
Place a piece of the toasted, buttered French bread into the bottom of each pre-heated soup dish. Ladle the hot soup over the toast slices. Sprinkle with additional Parmesan cheese, if desired. Serve hot.