In a medium heatproof mixing bowl, whisk together 1 whole large egg, 2 large egg yolks, 1/2 cup granulated sugar, 3 Tbsp cornstarch, and 1/8 tsp fine sea salt until completely smooth and slightly lightened in color, about 1–2 minutes.
In a large saucepan, combine 1 cup whole milk, 1 cup heavy whipping cream, and 2 Tbsp unsalted butter. Heat over medium, stirring occasionally, until the mixture is steaming and small bubbles form at the edges (just beginning to boil). Remove from heat as soon as it reaches this stage.
With the egg mixture whisking constantly, slowly drizzle in about one-third of the hot milk mixture in a thin stream to temper the eggs. Continue whisking until the mixture is smooth and warmed.
Pour the warmed egg mixture back into the saucepan with the remaining hot milk mixture and return the pan to medium heat.
Whisk constantly over medium heat until the mixture thickens and becomes smooth and glossy, about 1–3 minutes. As soon as the custard reaches a gentle simmer (small steady bubbles), reduce the heat to low and continue whisking vigorously for 1 full minute to cook the cornstarch.
Remove the saucepan from the heat and whisk in 2 tsp vanilla extract until incorporated.
Transfer the pastry cream to a shallow bowl. Press a piece of plastic wrap directly onto the surface of the hot cream to prevent a skin from forming. Let come to room temperature, then refrigerate until fully chilled, at least 2 hours. The pastry cream will keep in the refrigerator up to 3 days.