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1
Bring cornmeal, water, and 2 1/4 teaspoons salt to a boil in a 4-quart heavy saucepan, stirring constantly.
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2
Reduce heat and simmer, stirring frequently, until polenta is creamy and tender, about 50 minutes.
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3
Remove from heat, then stir in 11/2 tablespoons butter and cool slightly.
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4
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).
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5
Cover with plastic wrap, then poke several holes in wrap with a small sharp knife and chill 2 hours.
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6
Melt remaining tablespoon butter in a heavy saucepan over moderate heat.
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7
Stir in sugar, cumin, cinnamon, paprika, cayenne, and 1/4 teaspoon salt, then cook, without stirring, until caramelized.
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8
Add pumpkin seeds and cook, stirring occasionally, until seeds are puffed and golden.
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9
Transfer to a plate to cool.
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10
When seeds have hardened, break up any clumps with your fingers.
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11
Whisk together pomegranate juice, vinegar, and shallot and let stand 5 minutes.
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12
Whisk in 3 tablespoons oil, remaining 1/4 teaspoon salt, and pepper to taste.
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13
Preheat oven to 450F.
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14
Cut pumpkin quarters crosswise into 1/2-inch-thick slices.
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15
Toss with 1 tablespoon oil and salt and pepper to taste in a shallow baking pan and arrange slices in 1 layer.
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16
Roast in middle of oven until just tender, about 20 minutes.
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17
Remove from oven, then cover with foil to keep warm.
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18
Trim polenta into a 9- by 6-inch rectangle.
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19
Cut polenta into 6 (3-inch) squares, then halve each square diagonally.
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20
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).
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21
Transfer as cooked to a plate and keep warm, covered.
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22
Shave 12 strips from cheese with a vegetable peeler.
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23
Whisk vinaigrette, then toss arugula in a large bowl with enough vinaigrette to lightly coat.
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24
Place several pieces of pumpkin and 1 piece of polenta on each of 6 plates.
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25
Top with arugula, more pumpkin, and remaining polenta.
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26
Sprinkle with candied pumpkin seeds and top with parmesan shavings, then drizzle with remaining vinaigrette.
