Using a mandolin slicer (preferred) or a knife, cut cucumber into very thin slices.
Slice onion slightly thicker (approximately 1/16-inch thicker). Cut round onion in half and separate.
Mix brine ingredients, stirring until sugar and salt are dissolved.
Place cucumbers and onions in a large nonreactive bowl (ceramic, glass or stainless steel).
Pour brine over cucumber mixture and stir to make sure it is immersed in the brine.
Cover and refrigerate at least 24 hours.