Preheat oven to 500°F. Toss together brussels sprouts, oil, and ¼ teaspoon salt on a large rimmed baking sheet. Spread in a single layer, and roast in preheated oven until tender and lightly charred, about 12 minutes. Transfer to a plate to cool. Reduce oven temperature to 300°F.

Spread hazelnuts on a large rimmed baking sheet, and bake at 300°F until golden and skins wrinkle, about 8 minutes. Transfer to a clean kitchen towel, and rub off skins. Let stand until cool enough to handle; coarsely chop.

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 salt.

To serve, spread tahini mixture on a platter, and top with remaining brussels sprouts. Garnish with hazelnuts, chopped parsley, and a generous drizzle of olive oil.