Slice the cucumbers as thin as possible, using a mandolin if desired. Peel and thinly slice the onion. Layer the cucumbers and onions in a large shallow bowl, sprinkling salt between each layer and alternating onion layers between cucumbers. Let stand for about 45 minutes to 1 hour, then drain thoroughly.
Transfer to a serving bowl. Add 1/2 cup white vinegar, 1 to 2 tablespoons sugar, fresh ground pepper and additional salt, if desired. Taste and adjust seasonings. Stir in a few ice cubes and let stand 10 to 15 minutes, stirring occasionally.