Active time: 1 3/4 hr Start to finish: 5 hr
Makes 6 servings.
Bring cornmeal, water, and 2 1/4 teaspoons salt to a boil in a 4-quart heavy saucepan, stirring constantly. Reduce heat and simmer, stirring frequently, until polenta is creamy and tender, about 50 minutes. Remove from heat, then stir in 11/2 tablespoons butter and cool slightly.
Spoon polenta onto center of a lightly buttered large baking sheet, then spread evenly into a 10- by 7-inch rectangle (about 1/2 inch thick). Cover with plastic wrap, then poke several holes in wrap with a small sharp knife and chill 2 hours.
Candy pumpkin seeds:
Melt remaining tablespoon butter in a heavy saucepan over moderate heat. Stir in sugar, cumin, cinnamon, paprika, cayenne, and 1/4 teaspoon salt, then cook, without stirring, until caramelized. Add pumpkin seeds and cook, stirring occasionally, until seeds are puffed and golden. Transfer to a plate to cool. When seeds have hardened, break up any clumps with your fingers.
Whisk together pomegranate juice, vinegar, and shallot and let stand 5 minutes. Whisk in 3 tablespoons oil, remaining 1/4 teaspoon salt, and pepper to taste.
Preheat oven to 450°F.
Cut pumpkin quarters crosswise into 1/2-inch-thick slices. Toss with 1 tablespoon oil and salt and pepper to taste in a shallow baking pan and arrange slices in 1 layer. Roast in middle of oven until just tender, about 20 minutes. Remove from oven, then cover with foil to keep warm.
Fry polenta while pumpkin roasts:
Trim polenta into a 9- by 6-inch rectangle. Cut polenta into 6 (3-inch) squares, then halve each square diagonally. Heat 1 tablespoon oil in a large nonstick skillet until hot but not smoking, then cook polenta in 2 batches, turning once, until golden brown, about 8 minutes (if necessary, use remaining tablespoon oil for second batch). Transfer as cookedto a plate and keep warm, covered.
Shave 12 strips from cheese with a vegetable peeler.
Whisk vinaigrette, then toss arugula in a large bowl with enough vinaigrette to lightly coat. Place several pieces of pumpkin and 1 piece of polenta on each of 6 plates. Top with arugula, more pumpkin, and remaining polenta. Sprinkle with candied pumpkin seeds and top with parmesan shavings, then drizzle with remaining vinaigrette.
• Polenta, spread on baking sheet and not yet fried, can be chilled up to 1 day.
• Candied pumpkin seeds can be prepared 3 days ahead and kept in an airtight container at room temperature.
• To juice a pomegranate, firmly roll it on a work surface until it feels softer, then cut a small hole in skin and squeeze.