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In a medium bowl, mix together ¼ cup of the milk and the egg yolks. Set aside.
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In a medium saucepan, add the remaining milk, sugar, cornstarch and salt over medium heat. Stirring often, bring to a light simmer.
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Add ½ cup of the warm milk mixture to the egg yolks, whisking to combine.
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Add another ½ cup of the warm milk to the egg mixture another two times (three times total). This tempers the egg yolks so they don’t curdle.
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Add this mixture to the remaining milk in the saucepan.
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Lower the heat to medium-low and continue heating another 4-6 minutes, stirring with a spatula, until it is thick enough that it covers the back of the spatula.
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Remove from heat and stir in butter and vanilla, stirring until butter is melted.
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Cover with plastic wrap, pressing it directly on the surface of the pudding to avoid it developing a skin, and let cool completely.