Fill a large bowl with ice water, then set a fine-mesh sieve over a medium bowl. Set both aside.
In a 3 to 4-quart saucepan, whisk together the sugar, cornstarch and salt. Whisk in the egg yolks until thoroughly combined and all of the dry ingredients have been incorporated. Then, whisk in the milk to combine.
Place the saucepan over medium heat, whisking constantly, until the pastry cream begins to noticeably thicken, about 4 to 8 minutes. When it reaches this point, look for large bubbles to break on the surface. Once they do, whisk constantly for 1 minute, then remove from the heat.
Whisk in the butter and vanilla until melted and completely smooth.
Strain the pastry cream through the fine-mesh sieve into the medium bowl. Immediately press a piece of plastic wrap against the surface of the pastry cream to prevent a skin from forming. Place the bowl of pastry cream into the ice bath to cool for 30 minutes. Transfer to the refrigerator and chill for at least 4 hours or overnight.