In a medium saucepan, combine the 1 cup of sugar and water. Cook over medium heat, without stirring, until the sugar dissolves and turns into a deep amber color. This usually takes about 5-7 minutes. Watch it closely to prevent burning!
Carefully remove the saucepan from the heat. Slowly pour in the warmed heavy cream, being careful as it may splatter. Add the butter and whisk until smooth. Stir in the sea salt. Set aside about 1/4 cup of this salted caramel sauce for topping later.
In a separate bowl, whisk together the cocoa powder and cornstarch until well combined. This step is crucial to prevent lumps in your pudding.
In a medium saucepan, whisk together the milk and the additional 1/2 cup of sugar. Add the cocoa powder and cornstarch mixture. Cook over medium heat, whisking constantly, until the mixture comes to a simmer and begins to thicken.
Reduce the heat to low and continue to cook, whisking constantly, for about 2-3 minutes, or until the pudding is smooth and thickened. Remove from heat and stir in the remaining salted caramel sauce and vanilla extract.
Pour the Salted Caramel Chocolate Pudding into individual serving dishes or a large bowl. Cover with plastic wrap, pressing it directly onto the surface of the pudding to prevent a skin from forming. Refrigerate for at least 2 hours, or until completely chilled.