Wash lemon well in hot water. Using a vegetable peeler, peel off 4 strips of zest, about 1 1/2 inches long. Make sure that it is thin enough that it has very little pith (white). Set aside.
Sprinkle each of the sugar cubes with 3 dashes of bitters each, and place in the bottom of a Champagne flute. Fill each glass with Champagne, being careful to pour slowly to avoid it bubbling over. Garnish each flute with one of the lemon twists and serve immediately.