Preheat the oven to 450 degrees F with a rack in the upper third of the oven. Line a large, rimmed baking sheet with parchment paper or aluminum foil. Halve the eggplants lengthwise and brush the cut sides all over with a thin layer olive oil. Place them in the prepared pan with the halved sides down.
Roast the eggplant until the flesh is very tender, about 35 to 40 minutes (this might take more or less time depending on the size of your eggplant). You can tell when the eggplant are ready by gently pressing on the skin; they should deflate slightly. Cool for a few minutes, then flip the eggplant over with a spatula and scoop out the flesh with a large spoon. Discard the skins.
Place a colander or mesh strainer over a mixing bowl, then transfer the eggplant to the strainer. Let the eggplant rest for a few minutes, occasionally stirring and pressing lightly to remove as much excess water as possible. Discard the liquid and transfer the eggplant to a medium-sized bowl.
Mash with a fork until the desired texture is reached (baba ganoush typically has some texture). Stir in 3 tablespoons of olive oil as well as the lemon juice, garlic, tahini, salt, pepper, and paprika until evenly combined.
Transfer to a serving bowl and garnish with parsley and a light sprinkle of smoked paprika. You can also drizzle on a bit more olive oil for presentation.
Serve with your favorite accompaniments. Store leftovers in an airtight container in the refrigerator for up to 4 days.