In a large saucepan over medium-low heat, combine light brown sugar, water, and salt. Swirl gently until boiling, about 5-6 minutes.

While caramel cools, whisk together half and half, egg yolks, and cornstarch in a mixing bowl until smooth.

Gradually mix the dairy mixture into the cooled caramel, whisking constantly to combine.

Return to heat, whisk gently as it thickens for about 5-7 minutes until it coats the back of a spoon.

Remove from heat, stir in butter, dark rum (if using), and vanilla extract.

Strain the pudding mixture through a fine-mesh sieve into ramekins or serving dishes.

Allow to cool at room temperature, then refrigerate until fully set for at least 2 hours.