Prepare the eggplant: Arrange the eggplant slices on paper towels and salt generously on both sides. Let the eggplants rest for about an hour to expunge their excess liquid. Pat completely dry with more paper towels and set aside.

Fry the eggplant: Heat the oil in a large skillet over medium-high heat. Working in batches, add the eggplant slices and fry until deeply browned on both sides, about 2 to 3 minutes per side. Transfer to a paper towel-lined baking sheet and repeat with the remaining eggplant. Cover loosely with foil and set aside.

For the eggplant in paprika: Combine the sweet and hot paprikas in a small bowl. Sprinkle one-third of the paprika mixture over the bottom of an 11-inch by 7-inch baking pan. Top with half of the garlic slices. Arrange one-fourth of the warm fried eggplant in an even layer over top of the paprika and garlic. Sprinkle with 1 tablespoon of the vinegar. Repeat the layering process and sprinkle the top of the eggplant with the final third of the paprika mixture. Let rest for about an hour at room temperature before serving in the baking dish.

For the eggplant in vinegar: Combine the remaining eggplant slices in a large bowl with the onion, garlic, allspice, and bay leaves. Whisk the vinegar and oil in a small measuring cup and season with salt and pepper to taste. Pour the vinaigrette over the eggplant mixture and toss gently to coat. Cover the surface directly with plastic wrap, pressing lightly to compress the ingredients. Let rest for about an hour at room temperature.

To serve: Using a slotted spoon or tongs, transfer the eggplant and onion slices to a rimmed serving dish. Drizzle with some of the vinaigrette and serve.