In a medium saucepan, combine light brown sugar, water, and a pinch of salt over medium-low heat. Stir gently until it comes to a boil, then simmer for about 5-6 minutes or until deep amber.

In a mixing bowl, whisk together half and half, egg yolks, and cornstarch until smooth. Set aside.

Slowly add the dairy mixture into the cooled caramel, whisking constantly. Pour it back into the saucepan and whisk until it thickens and bubbles, about 3-5 minutes.

Once thickened, remove from heat and stir in butter, dark rum (if using), and vanilla extract. Strain for a silky texture.

Pour the budino into ramekins, cool at room temperature, then chill in the fridge for at least 2 hours. Drizzle with caramel and sprinkle with salt before serving.