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Pour enough oil to coat the center of a baking sheet. Season the scored eggplant with salt and pepper, then arrange the cut sides down in the EVOO. Drizzle the garlic with EVOO and season with salt and pepper. Wrap in a foil pouch and roast along with the eggplant until very tender, 40 minutes. Cool to handle.
Heat 2 tablespoons EVOO, a couple of turns of the pan, in a Dutch oven over medium-high heat. Add the thyme, crushed red pepper flakes, onions and salt and pepper and stir to soften, 5 minutes. Add the tomato paste and stir 1 minute more. Then add the stock, wine, and tomatoes. Scrape the eggplant away from skin and add the flesh, along with the garlic, squishing the cloves from the skins. Add the basil. Puree the soup with an immersion blender. Simmer to thicken. Cool and store for a make-ahead meal. Reheat the soup over medium heat, and top with the basil and cheese.