Trim and peel the eggplant, then cut it lengthwise into 1 cm thick slabs. Season all over with salt and let sit for 20 minutes. Press with paper towels to absorb moisture, then cut into 2.5 cm pieces. Meanwhile, halve, peel and thinly slice the onion lengthwise. Heat 2 tbsp olive oil in a large saucepan over medium-high heat and cook the onion, stirring occasionally, until lightly charred at the edges but not fully caramelized, 10 to 15 minutes. Transfer to a bowl. Add the remaining 3 tbsp oil and fry the eggplant, flipping once, until golden brown on both sides, about 8 minutes. Drain on paper towels. Return onion and eggplant to the pan and grate in the garlic. Cook, stirring often, until soft and the garlic smells sweet, 5 to 10 minutes. Remove from the heat and let cool. Stir in the labneh and season with salt. Transfer to a serving dish, cover and refrigerate for at least 2 hours.