In a medium saucepan, stir together 1¾ cups of the milk, the sugar, chocolate and salt, and place over medium-low heat. Cook, stirring occasionally, until the chocolate is completely melted.
In a small bowl, whisk together the remaining ¼ cup milk and cornstarch until no lumps remain. Pour the cornstarch mixture into the saucepan and whisk to combine. Cook, stirring frequently, until the mixture thickens and begins to boil. Reduce the heat to low and continue to cook for an additional minute, whisking constantly. Remove from the heat and whisk in the vanilla extract.
Pour the pudding mixture into a clean bowl (or individual serving cups or ramekins), and press a piece of plastic wrap against the surface of the pudding to prevent a skin from forming. Refrigerate the pudding until completely chilled, at least 4 hours. Whisk prior to serving, if storing in a large bowl. The pudding can be kept in the refrigerator, covered, for up to 4 days.
Storing: Once the chocolate pudding cools, you do not need to keep the plastic wrap on the top layer of the pudding. Simply store in an airtight container in the refrigerator.
Shelf Life: Stored properly in the refrigerator, this homemade pudding should last up to 4 days.
Freezing: I would not recommend freezing and re-thawing this pudding recipe. Alternatively, you can freeze this pudding in a popsicle mold for a delicious Chocolate Pudding Pop!