1 1/2 pounds large Italian eggplants, peeled and cut crosswise into 1-inch-thick rounds Kosher salt
2/3 cup vegetable oil
2 large red peppers, thinly sliced
1 large Spanish onion, thinly sliced
3 garlic cloves, thinly sliced
1 tablespoon sweet smoked paprika
1/4 cup sherry vinegar
1/2 cup chopped parsley
2 tablespoons fresh lemon juice
Line a large rimmed baking sheet with paper towels. Arrange the eggplant slices in a single layer on the paper towels and sprinkle them with salt. Let stand for 1 hour. Pat the eggplant dry with paper towels.
In a large skillet, heat 1/3 cup of the vegetable oil until shimmering. Add half of the eggplant and cook over moderately high heat, turning once, until browned and tender, about 8 minutes. Using a slotted spoon, transfer the eggplant to fresh paper towels to drain. Repeat with the remaining 1/3 cup of vegetable oil and eggplant.
Add the peppers, onion and garlic to the skillet and cook over moderate heat, stirring occasionally, until softened, about 10 minutes. Stir in the paprika and cook until fragrant, about 1 minute.
Return the eggplant to the skillet, stir gently to coat and cook for 5 minutes. Stir in the sherry vinegar and simmer for 1 minute. Remove from the heat. Stir in the parsley and lemon juice and season with salt. Let cool to room temperature, then refrigerate until lightly chilled, about 20 minutes, before serving.