Thinly slice and salt the eggplants. Let them weep for 20 minutes, then pat dry with paper towel. Put the oil in a large, heavy-bottomed skillet and bring it to the just-smoking point. Put in one layer of eggplant slices. Flip them over when the bottoms have turned a dark-reddish golden brown. Fry them on the other side. Drain them well on paper towels, covering both the top and bottom of the eggplants. Fry the rest in batches, following the same procedure. Lightly salt the finished product and sprinkle with one-half of the lemon juice.
Add the remaining lemon juice to the yogurt, along with the garlic, the cumin, the olive oil, and salt and pepper. Mix well with a fork.
Arrange the eggplant slices in a shallow bowl: first a layer of slices, drizzled with the yogurt mixture, followed by another layer drizzled with more yogurt mixture, and so on. Drizzle all remaining sauce over the top of the entire dish and drift with paprika--or even with more cayenne pepper if you want it spicier.