Preheat oven to 350°. Combine red onion, sugar, a big pinch of salt, 2 Tbsp. vinegar, and 2 Tbsp. hot water in a small bowl. Let sit, tossing occasionally, until you’re ready to assemble sandwiches.
Brush eggplant rounds generously with oil on one side. Heat a large skillet, preferably nonstick, over medium-high. Working in batches if needed, cook eggplant in skillet, in a single layer, until well browned, about 4 minutes. Brush each round with oil, turn over, and cook until other side is browned and eggplant is tender, about 3 minutes. Transfer to a baking dish or wide shallow bowl.
Whisk 1 Tbsp. plus 1½ tsp. oil and remaining 1 Tbsp. red wine vinegar in another small bowl to combine; season with salt and pepper. Pour over eggplant, toss gently to coat, and set aside.
Toast walnuts on a rimmed baking sheet, tossing halfway through, until golden brown, 6–8 minutes. Let cool slightly, then pile on a cutting board along with basil. Add a pinch of salt and finely chop. Transfer to a clean small bowl and add ricotta. Finely grate in garlic and mix well; season with salt and pepper.
Spread ricotta mixture over toast. Top with lettuce leaves, pickled onion, and reserved eggplant. Add grilled hauloumi cheese.
Serve & enjoy!