In a medium saucepan set over medium-high heat, combine the milk, brown sugar, cornstarch, and salt. Cook, stirring constantly, until the mixture darkens and starts to thicken, about 5-8 minutes.
2½ cups whole milk, 1 cup brown sugar, ⅓ cup cornstarch, ½ teaspoon kosher salt
Reduce the heat to medium-low. Whisk in the egg yolks, and continue to cook and stir until the filling is thickened.
3 egg yolks
Remove the saucepan from the heat and stir in the butter and vanilla.
3 tablespoons unsalted butter, 1½ teaspoons pure vanilla extract
Pour the filling into the prebaked pie crust.
1 9-inch refrigerated pie crust
Press a piece of plastic to the surface of the pudding and set the pie aside while you make the meringue.
Preheat the oven to 350°F.