Whisk 1/3 of the cold milk and cornstarch together in a medium bowl until smooth.

Pour remaining milk and sugar into a small saucepan. Heat over medium heat, stirring occasionally, until milk begins to simmer; remove from heat and set aside.

Stir egg and egg yolks into cornstarch mixture; gradually whisk in hot milk. Pour mixture back into the small saucepan; cook over medium-low heat, whisking constantly, until thickened and mixture coats the back of a spoon, about 3 to 5 minutes.

Remove custard from heat; whisk in butter and vanilla until smooth.

Transfer into a bowl and cover directly with plastic wrap to prevent a skin forming; allow to cool before using.