In a large bowl, whisk together sugar, cornstarch, and salt. Whisk in eggs and egg yolks until well combined, stopping to scrape sides of bowl.

In a medium saucepan, heat milk over medium heat, stirring frequently, until steaming. (Do not boil.) Gradually whisk warm milk into sugar mixture. Return mixture to saucepan; cook over medium heat, whisking constantly, until thickened, slightly lumpy, and beginning to bubble, about 4 minutes. Cook, whisking constantly, for 2 minutes. Remove from heat. Whisk in vanilla bean paste. (Mixture should now be smooth again.) Pour into a large rimmed baking dish, spreading into a thin layer. Cover with a piece of plastic wrap, pressing wrap directly onto surface of custard to prevent a skin from forming. Refrigerate until thoroughly chilled, about 1 hour.

Let custard stand at room temperature for 40 minutes; stir until smooth and softened.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter at medium speed until light, smooth, and creamy, about 4 minutes, stopping to scrape sides of bowl. Add custard, 1 to 2 tablespoons (16 to 32 grams) at a time, beating until combined after each addition and stopping to scrape sides of bowl.

Switch to the whisk attachment. Beat at medium speed until light and fluffy, 3 to 4 minutes. Use immediately, or refrigerate in an airtight container for up to 1 week. If refrigerating, let come to room temperature and rewhip until fluffy and light before using.