Whisk milk, sugar, and cornstarch in a large saucepan; bring to a boil. Boil 3 minutes, whisking constantly.
Beat eggs with a whisk in a large bowl. Gradually add half the hot milk mixture to the beaten eggs. Return milk-egg mixture to pan. Cook over medium heat 3 minutes, or until thick, whisking constantly. Remove from heat.
Stir in pumpkin, salt, and pumpkin-pie spice. Spoon evenly into 6 (6 oz.) custard cups.
Let cool, and chill for about 30 minutes, or until pudding is set.
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