How to Make It
Brine the eggplant In a large bowl, mix 8 cups of warm water with the salt and sugar, stirring to dissolve. Cut a 1/8-inch-deep slit lengthwise down one side of each eggplant piece; do not cut in half. Add the eggplant to the brine and cover with a plate to submerge. Cover with plastic wrap and refrigerate for at least 3 hours or, preferably, overnight. Drain well and discard the brine.
Meanwhile, prepare the onions In a small bowl, soak the onions in ice water for 15 minutes. Drain. In the same bowl, mix ½ cup of warm water with the vinegar, salt and sugar, stirring to dissolve the salt and sugar. Add the soaked onions. Cover and refrigerate for at least 3 hours or overnight.
Make the spice mixture In a small, dry skillet, toast all of the spices over moderately high heat until fragrant, about 2 minutes. Transfer to a spice grinder and let cool, then grind to a powder.
Finish and assemble the dish In a small skillet, cook the olive oil and garlic over moderately high heat, stirring occasionally, until the oil is hot and fragrant, about 3 minutes. Remove the skillet from the heat and let the garlic oil cool. Meanwhile, in a medium bowl, toss the cherries with the sugar, lemon juice and a pinch of salt.
In a large cast-iron skillet, heat 2 inches of canola oil to 325°. Add the drained eggplant and fry in batches, turning occasionally, until the eggplant is completely tender, about 5 minutes per batch. Using a slotted spoon, transfer the eggplant to a paper towel–lined baking sheet to drain. When cool enough to handle, open up the eggplant pieces along the cut side so that they are flattened but still intact. Sprinkle the eggplant all over with the spice mixture and season all over with salt.
Carefully pour off all but 1 tablespoon of the hot canola oil from the skillet. Heat the skillet over moderately high heat and cook half of the eggplant, turning once, until golden brown, about 4 minutes. Transfer to a plate and brush with the garlic oil. Repeat with the remaining eggplant.
Just before serving, drain the onions. Transfer to a bowl and toss with the toasted hazelnuts, parsley and tarragon. Spread the lebneh on 4 plates and top with the eggplant, cherries and the onion-herb mixture. Top with grated lemon zest and grated hazelnuts and serve.