How to Make It
In a saucepan, combine the water, vinegar, sugar and bay leaf; bring to a boil. Add the eggplant and onion and cook over moderately low heat, stirring occasionally, until the eggplant is tender but not falling apart, 20 minutes. Drain the eggplant and return it to the saucepan. Stir in the butter; season with salt and pepper. Discard the bay leaf.
Meanwhile, in another saucepan, heat 1/4 cup of the olive oil. Add the tomatoes, garlic, thyme and lemon juice and season with salt and pepper. Cook over moderate heat until thickened, 20 minutes. Discard the thyme sprigs.
Preheat the broiler. Season the snapper with salt and pepper. In a large nonstick ovenproof skillet, heat the remaining 2 tablespoons of olive oil until shimmering. Add the snapper, skin side down, and cook over moderately high heat until browned, 5 minutes. Transfer the skillet to the broiler and broil the fish until just cooked through, about 5 minutes longer. Spoon the eggplant onto plates, top with the snapper and serve with the tomato sauce on the side.