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Whisk together the sugar, cornstarch, and salt in a medium sauce pan that is not heated.
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Very gradually (a few tablespoons at a time) whisk in milk. Carefully dissolve the cornstarch.
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Whisk in eggs and egg yolk.
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Over medium heat and whisking constantly cook until the first set of bubbles form and break.
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Reduce heat to low; still whisking, cook 2-3 minutes.
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Remove from heat; immediately pour through a fine-mesh sieve into bowl to remove any clumps. Stir butter and vanilla into hot pudding. Mix well.
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Place plastic wrap directly on surface of pudding and around the bowl to prevent that rough skin from forming.
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Place in the refrigerator and chill at least 3 hours, if not overnight.