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Melt chocolate in a saucepan or in the microwave and allow to cool slightly.
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Sift cocoa, corn starch, sugar and salt together in a large, heavy-bottomed saucepan.
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Slowly whisk in the half and half, then the egg yolks, then the milk.
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Stir in the chocolate (it will look curdled and clumped, but will melt together with cooking).
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Bring mixture to a boil over medium-high heat, whisking constantly and scraping the sides and bottom of the pot.
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A splatter screen might be helpful here if your pot is not very deep.
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As the pudding reaches the boiling point it will darken and thicken.
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When it boils, reduce heat to medium and stir with a wooden spoon for about two minutes, until it has reached the appropriate pudding consistency.
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Stir in butter and vanilla.
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Serve warm or chilled.