Cut the eggplant lengthwise in half and score the flesh in a crosshatch pattern to allow more even cooking. Place flesh-side up in a baking dish along with the garlic cloves. Drizzle everything with olive oil and sprinkle with salt and pepper.
Roast until the flesh is very tender, 25 to 30 minutes. Remove and let sit until cool enough to handle.
Scoop out the soft flesh into the bowl of a food processor. Add the roasted garlic, sour cream, lemon juice, and paprika.
Blend until smooth and creamy, scraping down the sides of the bowl as needed. Taste and adjust seasonings as desired.
Transfer to a serving dish and swirl a spoon through the top of the dip. Drizzle with the remaining olive oil and sprinkle with a little more paprika.