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Set a fine mesh sieve over a medium bowl and set it aside.
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In a medium saucepan, heat the milk over medium just to until bubbles start to form around the edges of the pan.
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Meanwhile, in a large bowl, whisk together the sugar, cornstarch, and salt until fully combined and free of lumps. Add the egg yolks and whisk until the mixture is pale yellow and thick, about 1 minute.
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Gradually whisk about half of the hot milk into the egg mixture, then pour the mixture back into the saucepan, whisking to combine.
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Whisking constantly, cook over medium heat until thickened and bubbles are forming and bursting in the center of the saucepan. Once you reach this point, continue whisking for 1 additional minute.
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Use a rubber spatula to press the mixture through the fine mesh sieve into the bowl. Add the vanilla and stir to combine. Add the butter a few pieces at a time, stirring until melted and fully incorporated after each addition.
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Cover the pastry cream with a piece of plastic wrap or parchment paper pressed directly to the surface and refrigerate until completely cool, at least 4 hours. It will thicken significantly as it cools.
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When ready to use the pastry cream, add it to a large bowl or the bowl of a stand mixer fitted with the flat beater. Beat on medium-high speed until smooth and glossy, 30 to 60 seconds.
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Store pastry cream in the refrigerator, covered, for up to 5 days. Do not freeze.
