Heat broiler with rack 6 inches below heating element. Drizzle eggplants generously with oil; season with salt. Place on a rimmed baking sheet, cut-side down. Broil until skins are blistered, about 10 minutes. Flip and broil until cut sides are blackened and eggplants are very soft, about 8 minutes more.
Meanwhile, cook pasta in a large pot of generously salted boiling water until al dente. Reserve 1 cup pasta water; drain pasta.
Scoop flesh from eggplants, discarding skins; transfer to a large bowl. Mash with oil, garlic, lemon juice, and tahini. Stir in pasta, 1/2 cup reserved pasta water, and cilantro. Season with salt and pepper. Add more pasta water, a little at a time, until sauce evenly coats pasta. Serve, with yogurt and cilantro sprigs, and drizzled with oil.