Rinse the eel pieces under cold water and pat them dry with paper towels.
In a large pot, melt the butter over medium heat.
Add the finely chopped onion and minced garlic, and sauté until the onion is soft and translucent.
Pour in the fish broth and dry white wine. Bring the mixture to a simmer.
Add the eel pieces to the pot. Cover and simmer gently for about 15-20 minutes, or until the eel is cooked through and tender.
While the eel is cooking, combine the chopped parsley, sorrel (or spinach and lemon juice), chervil, watercress, tarragon, mint, and dill in a bowl.
After the eel is cooked, add the chopped herbs to the pot. Stir gently to combine.
Season with ground black pepper and salt to taste.
Continue to simmer for another 5 minutes to allow the flavors to meld together.
Transfer the eel and green herb sauce to a serving dish.
Enjoy your meal!