Turn one gas burner on medium-high and place the eggplant directly over the flame. Use a pair of stainless steel tongs to turn the eggplant every 5 minutes or so, until its skin is charred and crispy on all sides (15 to 18 minutes). When it's done, the eggplant should be really soft. If you don't have a gas burner, you can use a grill or roast it in the oven (see notes).
Remove the eggplant to a bowl and let it cool down.
When the eggplant is cool enough to handle, peel off its charred and crispy skin - it should come right off. Get rid of the skin, as well as the stem, but don't worry if a few bits of the skin remain - they add extra flavor!
Transfer the peeled eggplant to a strainer over a bowl and let it stand 30 minutes. Alternatively, you can place it in a salad bowl spinner and spin gently to speed up the process. Discard all drippings.
Place the eggplant pulp in a clean bowl and separate the strands with a fork.
Add tahini, Greek yogurt, lemon juice, olive oil, garlic, salt and cayenne pepper. Stir vigorously with your fork or a wooden spoon until the mixture is pale and creamy. If you prefer a smoother texture, this step can be done in a food processor.
Transfer to a serving bowl, cover and refrigerate for 30 minutes to let the flavors combine. Serve with a drizzle of olive oil, and a few flakes of Aleppo pepper and parsley, to taste.