In a medium saucepan, use a wooden spoon or silicone spatula to stir together the sugar, cocoa, instant coffee granules, cornstarch, and salt.
Slowly stir in the milk and keep stirring until smooth and combined over low heat.
Stir the mixture constantly over medium-low heat, scraping the bottom, sides, and corners of the pan.
Continue until the pudding base thickens.
Once it begins to slightly bubble, coat the back of the spoon or spatula to check for doneness. If it coats, then the pudding base is ready.
If the pudding is coating the bottom of the pan too quickly, reduce the heat.
Bring down the heat and add the chocolate. Stir vigorously until the pudding is very thick and smooth.
Take it off the heat and mix in the vanilla drops.
Serve warm, at room temperature, or chilled.