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Preheat oven to 375F. Cut off cauliflower ribs and leaves. Turn cauliflower upside down and cut florets off from center stem. (You can use ribs, leaves and stem for soup.) Divide cauliflower in medium florets. You should have about 1 3/4 pounds or about 10 cups florets.
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Line a roasting pan with foil if desired. Put cauliflower florets in roasting pan. Drizzle with 1 tablespoon olive oil and sprinkle lightly with salt. Roast for 20 minutes. Turn florets over using tongs, drizzle with 2 teaspoons olive oil and sprinkle with salt. Roast until just tender, about 15 more minutes. Remove 4 florets and reserve for garnish. Cover pan (or fold foil over remaining florets to cover them) and roast florets until very tender, about 10 more minutes. Remove cauliflower from pan and cool to room temperature.
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Meanwhile, make tahini sauce: Spoon tahini paste into a small bowl. Gradually stir in lemon juice and 4 tablespoons water. Add garlic and mix well. Add 1/4 teaspoon salt or to taste. Mixture should have consistency of thick mayonnaise.
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Transfer cauliflower to a food processor and blend to a puree. Add tahini sauce, 1 tablespoon olive oil and 2 tablespoons water and process mixture until well blended. Add 1 more tablespoon water if necessary so spread will not be dry. Transfer to a bowl. Taste and add more salt, more lemon juice and pepper if desired. Cover and refrigerate for 2 or 3 hours to blend flavors. Break reserved cauliflower florets into smaller pieces and reserve for garnish.
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At serving time, spread cauliflower mixture in a thick layer on a serving plate. With the back of a spoon, make a cavity in the center. Garnish with parsley and with reserved small cauliflower florets and drizzle center with olive oil. Sprinkle cauliflower mixture lightly with Aleppo pepper. Serve with fresh pita bread or other flatbread.