Preheat your oven to 400°F (200°C). Line two baking sheets with parchment paper.
Toss the eggplant cubes with 2 tablespoons olive oil, salt, and pepper. Spread them on one baking sheet.
On the second baking sheet, place the quartered tomatoes, chopped onion, and whole garlic cloves. Drizzle with the remaining olive oil and season with salt and pepper.
Roast both trays for 25-30 minutes, rotating halfway through. The vegetables should be tender and slightly caramelized.
Transfer the roasted vegetables to a large pot. Add the vegetable broth, tomato paste, thyme, and smoked paprika.
Simmer for 15-20 minutes on medium-low heat to allow the flavors to meld.
Using an immersion blender, carefully blend the soup until smooth. Alternatively, transfer to a regular blender in batches.
Taste and adjust seasoning as needed. Serve hot, garnished with fresh basil leaves.