Heat the olive oil in a Dutch oven over medium heat. Add the eggplant in two batches, cooking until golden and softened, about 5–6 minutes per batch. Transfer to a plate.
Add the mushrooms to the pot and cook for 5 minutes until they release their liquid and start to brown. Add the onion and carrot and cook another 7 minutes, stirring often. Stir in the garlic and cook 30 seconds, then add the tomato paste and cook 1–2 minutes.
Pour in the stock or wine and let it bubble for 2 minutes. Add thyme leaves, then crush the whole tomatoes into the pot with your hands. Return the eggplant, season with salt, pepper, and basil, and stir well. Cover and simmer for 20 minutes, stirring occasionally, until thick and rich.
Finish with the butter and cream, stirring until silky. Taste and adjust seasoning. Serve over pasta with Parmesan and, if you like, a dollop of ricotta.
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