To make the ice cream:
Place the half and half in a small pan and cook over low heat, whisking from time to time; until it is warm, about 175°F.
Place the egg yolks, brown sugar and salt in a small metal bowl and whisk until completely mixed.
Add ¼ cup of the warm half and half mixture to the eggs, whisking all the while. Continue adding half and half to the eggs, ¼ cup at a time, until you have added about 1½ cups. Slowly, whisking all the while, return the now half and half and egg mixture to the remaining milk mixture in the pan and continue cooking until it just begins to thicken or reaches about 185°F.
Do not allow the mixture to boil. Pour through a medium fine strainer into a metal bowl, discard the solids and set aside until it reaches room temperature.
Add the bananas and rum and mix well.
Cover and refrigerate until it reaches 40°F, about 3 hours. Transfer to an ice cream maker and proceed according to the manufacturer’s instructions.
To make the burnt caramel sauce:
Place the sugar and water in a pot and bring to a boil. Continue boiling, without stirring, until it begins to color on the edges, about 2 to 3 minutes. Cook, whisking all the while, until it turns a deep brown and just begins to smell burnt, 3 to 4 minutes.
Slowly and very carefully add the cream, continuing to whisk, until it is completely incorporated.
Off heat, add the vanilla and salt.
Serve immediately or cool to room temperature, cover and refrigerate.