Set a large saucepot over medium-low heat. Combine the brown sugar, water, and salt in the pot. Swirl the pot (don’t stir) and heat until boiling. Cook for 5-6 minutes, until a rich brown color is formed. Turn off the heat.
While the sugar is cooling, set out a mixing bowl. Mix the half and half, egg yolks, and cornstarch. Whisk well until smooth.
Add the dairy mixture to the slightly cooled sugar, whisking continually. The sugar will seize up at first, but will melt down as it cooks.
Place the pot back over medium-low heat. Continue to whisk as the mixture becomes smooth and then thickens into firm pudding consistency.
Remove from heat. Stir in the butter, rum, and vanilla extract. Then set a fine mesh strainer over a bowl and pour the budino into the strainer. Use a spatula to push it through, if needed.
Pour (or scoop) the budino into ½ cup ramekins or serving jars. Allow them to cool slightly, then top with thick caramel sauce and sea salt (if desired.) Cover and refrigerate until ready to serve.
The alcohol is completely optional, to add a little oomph. Feel free to skip it entirely, or use bourbon-flavored vanilla extract instead.Leftovers will keep well for up to a week stored covered in the refrigerator.