Heat broiler to high with the rack 6 inches from the flame. (Adjust for more space if the eggplant touches the broiler.) Prick the eggplant all over with a fork.
Broil the eggplant on a rimmed baking sheet until charred and soft, 20 minutes.
Remove from heat and cool. Scoop out the flesh from the eggplant and pulse it with the tahini, oil, lemon juice, garlic, salt, cumin, cayenne, and parsley in a food processor until smooth. Serve with the vegetables.
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