Prepare the Caramel: In a large saucepan, combine light brown sugar, water, and a pinch of salt over medium-low heat. Stir gently and allow it to simmer for 5-6 minutes until the mixture turns a rich amber color.

Mix the Cream Base: In a mixing bowl, whisk together half and half, large egg yolks, and cornstarch until smooth. Set aside.

Combine Caramel and Cream Mixture: Add the cooled caramel to the cream mixture, whisking continuously. Return it to the saucepan.

Cook the Budino Mixture: Place the saucepan on medium-low heat and continue whisking until it thickens to a firm pudding consistency, around 3-5 minutes.

Add the Finishing Touches: Remove from heat and stir in unsalted butter, dark rum (if using), and vanilla extract until fully incorporated.

Strain for Smoothness: Strain the mixture through a fine-mesh sieve into a large bowl to remove lumps.

Portion and Chill: Pour the strained budino mixture into ramekins, filling them three-quarters full. Allow to cool slightly before refrigerating until set, 2-3 hours.

Garnish and Serve: Drizzle with caramel sauce and sprinkle with sea salt before serving.