Cut eggplant into 1/2-in.-thick slices, score with a paring knife in a tic-tac-toe pattern on both sides, and season with about 1/2 tsp. salt. Let sit 10 minutes, then blot dry with paper towels.
Meanwhile, bring a large pot of water to a boil. Heat a large frying pan over high heat, 2 minutes. Cooking in 2 batches, swirl in oil (3 tbsp. per batch) and wait 1 minute. Add eggplant slices in a single layer. Once they start to color, turn slices over and cook until tender and golden, another 3 to 4 minutes. Transfer to a paper towel-lined baking sheet.
Chop eggplant coarsely and toss in a large bowl with paprika, cayenne, and 1 tsp. salt. Mince garlic clove, sprinkle with a pinch of salt, and mash to a paste with the flat side of your chef's knife. Add paste to eggplant along with cilantro.
Blanch tomatoes in boiling water, 30 seconds. Cool in a bowl of ice water, then slip off skins. Core tomatoes and cut each in half horizontally. Squeeze out seeds and pulp into a strainer set in a bowl. Chop tomatoes coarsely; add juice to bowl.
Heat a large (not nonstick) frying pan over medium heat, 2 minutes. Swirl in remaining 1 tbsp. oil, add rosemary and chile, and heat another minute. Add onion and sauté until onion is translucent, 3 to 4 minutes. Add chopped tomatoes and juices. Season with remaining 1/4 tsp. salt and cook until most of liquid has evaporated, 6 to 8 minutes. Add eggplant to pan, turn heat to high, and cook another 6 to 8 minutes, stirring often, until well integrated and jammy in texture. Serve at room temperature.
Lucques, Tavern, Larder at Maple Drive, A.O.C., all in Los Angeles