Combine 2 cups of the half-and-half with the sugar and salt in a medium saucepan and cook over medium- low heat, stirring occasionally, until the mixture just begins to steam, 2 to 3 minutes.
Whisk the cornstarch with the remaining 1/2 cup half-and-half in a bowl until completely smooth. Stir this into the hot half-and-half and cook, stirring occasionally, until the mixture thickens and just starts to simmer, another 3 to 5 minutes. Reduce the heat to very low and continue to cook, stirring constantly, until the pudding thickens noticeably and starts to stick to the pan, 3 to 5 minutes more. Add the butter and stir until it melts completely. Stir in the vanilla.
Pour the pudding into a large heatproof glass serving bowl, or 4 to 6 small ramekins or heatproof glasses. Put plastic wrap directly on top of the pudding to prevent the formation of a skin, or leave it uncovered if you like that. Refrigerate until chilled, at least an hour, and serve within a day.