Grate the cucumbers on the large holes of a box grater into a medium bowl. Wrap the grated cucumber in a piece of cheesecloth or a clean kitchen towel and squeeze the cucumber juice into a pitcher; discard the grated cucumber. Stir the lime zest and juice, elderflower liqueur and 5 or 6 dashes orange bitters into the cucumber juice. Refrigerate until ready to serve, at least 30 minutes.
Drop a sugar cube into each flute and add a few dashes of orange bitters to each. Pour the champagne into the pitcher and stir. Pour into the flutes.