Pour 2 ½ cups milk into a saucepan and place on the stovetop over medium-low heat.
While the milk is warming, mix together the sugar, cornstarch, and salt in a bowl.
Add the remaining ½ cup milk to the cornstarch mixture and whisk until the dry ingredients are well incorporated.
Gradually pour the cornstarch mixture into the saucepan of warm milk, whisking the whole time.
Cook the pudding, whisking constantly, until it reaches 175 F. (If you do not have a thermometer, cook until it visibly thickens and coats the back of a spoon.) Remove from heat.
Add the vanilla extract and butter, and stir until the butter melts.
Ladle the pudding into one large storage container or individual serving-size containers and place in the refrigerator for 1-2 hours, until chilled and set.
Serve chilled, garnished with whipped cream and/or vanilla wafer cookies, if desired.