
Preheat oven to 400°F. Toss together brussels sprouts, olive oil, and ¼ teaspoon salt on a large rimmed baking sheet. Spread in a single layer, and roast in preheated oven until lightly charred, about 15 minutes. Transfer to a plate to cool. Reduce oven temperature to 300°F.
Place tahini, lemon juice, cumin, ice cube, half of the cooled brussels sprouts, and remaining ¾ teaspoon salt in a food processor. Process until ice melts and a chunky paste forms. With processor running, add ½ cup cold water in a slow, steady stream, processing until mixture is emulsified and resembles the texture of hummus. Season to taste with additional salt.
To serve, spoon tahini mixture into a bowl, and top with remaining brussels sprouts. Garnish with hazelnuts, chopped parsley, and a drizzle of olive oil. Serve with warm pita.