Using a box grater, coarsely grate the cucumber lengthwise, stopping when you get to the seedy center. Transfer the cucumber to a bowl and stir in the vinegar, lemon juice, sugar and salt. Let stand for 15 minutes.
If using rye bread slices, use a 2-inch-round cookie cutter to cut out 36 rounds from the bread. Cover with a slightly damp towel.
Transfer the grated cucumber to a colander to drain; squeeze out the excess liquid. Return it to the bowl and stir in the cream cheese and scallion. Spread the cucumber cream cheese on the bread rounds, top each with a Persian cucumber slice and arrange on a platter. Sprinkle with pepper and serve.