1. Place the yogurt in a paper towel–lined sieve set over a large bowl—the bottom of the bowl should clear the bottom of the sieve by at least 2 inches. Place a saucer on top of the yogurt to weigh it down slightly. Let stand at room temperature for 1 hour. About 1 cup of the whey should drain off. Discard the whey. Place the drained yogurt in a medium bowl.

2. Meanwhile, peel the cucumber, split it lengthwise, and scoop out the seeds. Coarsely chop the cucumber. In a food processor fitted with the metal chopping blade, pulse the cucumber 8 to 10 times, until finely chopped but not pureed. Transfer to a medium bowl and toss with ½ teaspoon salt. Let stand for 1 hour. Drain into a wire sieve and rinse well under cold running water to remove the salt. Drain well. A handful at a time, squeeze the cucumber well to remove excess liquid, then stir the cucumber into the yogurt.

3. Stir in the scallion, dill, garlic, cumin, remaining 1/8 teaspoon salt, and cayenne. Cover and refrigerate for 1 hour for the flavors to blend. Serve slightly chilled.