Prepare the Caramel: In a large saucepan over medium-low heat, combine light brown sugar, water, and salt. Stir gently until the sugar dissolves, then increase the heat slightly. Allow the mixture to come to a boil, then watch as it turns a rich amber color, about 5-6 minutes.

Mix the Pudding Ingredients: In a separate mixing bowl, whisk together half and half, egg yolks, and cornstarch until smooth. Set aside while your caramel finishes cooling.

Combine Mixtures: Gradually pour the egg mixture into the slightly cooled caramel, whisking continuously to ensure it blends smoothly without curdling the eggs.

Thicken the Pudding: Return the saucepan to medium-low heat and cook the mixture, whisking frequently for about 5-7 minutes, or until it thickens to a velvety pudding consistency.

Finish with Flavor: Remove the saucepan from the heat and stir in the butter, dark rum (if using), and vanilla extract until well combined.

Strain and Cool: Strain the mixture through a fine mesh strainer into ramekins or jars to achieve a perfectly smooth texture.

Garnish and Refrigerate: Drizzle with thick caramel sauce and sprinkle a touch of coarse sea salt on top. Refrigerate for at least 2 hours before serving.