Preheat the oven to 425°F (220°C).
Place the quartered tomatoes, cubed eggplant, quartered onion, and garlic cloves on a large baking sheet.
Drizzle the olive oil evenly over the vegetables. Sprinkle with salt, black pepper, and dried thyme.
Toss the vegetables gently to coat them evenly with oil and seasonings.
Roast the vegetables in the preheated oven for 30 to 35 minutes, stirring halfway through, until they are tender and slightly caramelized.
Remove the roasted vegetables from the oven and let them cool for 5 minutes.
Transfer the roasted vegetables to a large pot. Add the vegetable broth, tomato paste, and smoked paprika.
Bring the mixture to a boil over medium-high heat, then reduce the heat to low and simmer for 10 minutes to blend the flavors.
Using an immersion blender, puree the soup until smooth. Alternatively, carefully transfer the soup in batches to a blender and blend until smooth, then return to the pot.
Stir in the chopped fresh basil and adjust seasoning with additional salt and pepper if needed.
Serve the soup hot, garnished with extra basil if desired.