If using whole-milk plain yogurt, drain in a paper-towel-lined sieve set over a bowl, covered and chilled, 8 to 12 hours. Mince garlic and mash to a paste with a pinch of salt, then transfer to a bowl. Whisk in lebneh or drained yogurt, 1/4 teaspoon salt, and enough water to make mixture resemble well-stirred sour cream. Prepare grill for cooking. If using a charcoal grill, open vents on bottom of grill, then light charcoal. Charcoal fire is medium-hot when you can hold your hand 5 inches above rack for 3 to 4 seconds. If using a gas grill, preheat burners on moderately high, covered, 10 minutes, then reduce heat to moderate. Brush eggplant slices generously with oil on both sides and sprinkle with remaining 3/4 teaspoon salt. Grill slices on lightly oiled grill rack, covered only if using a gas grill, turning over occasionally, until eggplant is tender, 12 to 15 minutes total. Transfer as cooked to a platter, arranging slices in 1 layer. Put a spoonful of yogurt mixture on each eggplant slice and top with a mint leaf.