Step 1 Start making the green onion puree about 30 minutes before cooking the fish. Cut the green onions in half crosswise, separating the white part from the green. Place the white halves in a medium saucepan with 3 tablespoons butter and 2 tablespoons water. Simmer very slowly until tender, about 10 minutes. Add half of the green portions (reserving the remaining green portions for garnish) and cook for one minute.
Step 2 Remove from the heat and puree in a blender or food processor. Pass the puree through a coarse strainer into a clean saucepan, pressing on the onions with the back of a rubber spatula. Discard the onion solids remaining in the strainer.
Step 3Rewarm gently as the fish is cooking, adding a little water to thin the sauce consistency, if necessary. Just before serving, stir in the remaining butter. Season with the vinegar and salt and pepper to taste. Makes one-half cup sauce.