Preheat broiler. Place eggplant on a foil-lined rimmed baking sheet and broil, turning occasionally, until soft and blistered all over, 15 minutes. Let cool 20 minutes, then cut open lengthwise and scoop out flesh into a fine-mesh sieve set over a bowl; press with a spatula to remove excess liquid.

Add eggplant to the bowl of a food processor with garlic, tahini, lemon juice, and 1 1/4 teaspoons salt. Process until smooth, 30 seconds. Drizzle in oil; pulse until combined.

Transfer mixture to a serving dish. Stir 2 teaspoons water into yogurt; drizzle over dip. Sprinkle with cayenne and red-pepper flakes. Serve with grissini or chips.