Put 2 cups milk (or half-and-half or heavy cream) and 1/4 cup of the granulated sugar in a heavy saucepan. Heat over medium, stirring occasionally, until the milk reaches a gentle simmer (small bubbles at the edge and steam rising). Do not let it boil.

While the milk heats, in a medium bowl whisk together 3 eggs, the remaining 3/4 cup granulated sugar, and 1/4 cup cornstarch until completely smooth and free of lumps.

When the milk is simmering, remove the pan briefly from the heat. Ladle out about 1/2 cup of the hot milk and very slowly whisk it into the egg mixture to temper the eggs (this warms the eggs so they won’t scramble).

With the saucepan back on medium-low, slowly pour the tempered egg mixture into the remaining milk in a thin, steady stream while whisking constantly.

Continue to cook over medium-low, whisking constantly, until the mixture thickens and comes to a gentle boil. Once it boils, keep whisking and cook for about 1 minute more to fully activate the cornstarch.

Remove the pan from the heat and immediately whisk in 2 tablespoons butter and 1 1/2 teaspoons vanilla extract until the butter is fully melted and the mixture is smooth.

Pour the pastry cream through a wire-mesh strainer into a heatproof, airtight container to remove any cooked egg bits or lumps.

Press a piece of plastic wrap directly onto the surface of the pastry cream to prevent a skin from forming.

Chill in the refrigerator until cold, at least 2 hours or overnight. Keep refrigerated and use within a few days.