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Put pistachios in the food processor and pulse until they are in small pieces. Add water and sugar and pulse until combined into a paste
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Transfer paste to a small saucepan over medium heat, add milk and whisk until steamy and hot
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Whisk egg yolks, sugar, cornstarch, and salt
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The mixture will be thick, but keep whisking until smooth
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You need to temper the eggs at this point by adding 1/2 cup of the steaming milk to the egg mixture and whisking to combine. Do it again, then add the entire mixture back into the pistachio-milk mixture
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Whisk constantly over medium heat until thick and bubbly
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Remove from heat and stir in butter and vanilla bean paste until butter is melted
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Pass pudding through a fine mesh strainer
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Transfer to a large bowl and cover with plastic wrap, pressing it onto the top of the pudding to prevent the weird skin situation
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Refrigerate overnight