In a large mixing bowl, whisk together the egg yolks, banana puree, sugar, cornstarch and salt until smooth and thoroughly combined.
In a medium saucepan heat the milk, cream, rum and vanilla over medium-high heat until steaming but not bubbling. Reduce the heat to the lowest setting.
While whisking the egg yolk mixture, gradually pour in about half of the hot milk mixture to temper the egg yolks. (I like to start with 2-3 tablespoons of hot milk first and whisking vigorously to prevent the eggs from curdling).
Pour the tempered egg mixture into the saucepan with the remaining milk mixture and cook over medium-low heat for 5-7 minutes until the mixture becomes thick and has a custard like texture (coats the back of a wooden spoon and drops off easily but isn't runny).
Remove the saucepan from heat and beat in the cold butter.
Pour the custard into a large bowl and allow to cool for a few minutes.
Cover the surface of the custard with plastic wrap to prevent a skin from forming, and chill in the refrigerator for 4 hours or up to two days (any longer and the bananas will turn the custard brown).
Divide mixture into four ramekins. Smooth the tops with the back of a spoon for an even surface.
Cover the surface of the custards with a generous layer of turbinado sugar. Lightly pat the sugar down with the back of a spoon.
Using a kitchen torch, broil the sugar for 30 seconds - 1 minute until the sugar is fully caramelized and resembles glass. Alternatively, you can set your oven to 500F (260 C) and broil the custards on the top rack for 2 minutes until caramelized and bubbly.
Serve immediately, or if making in advance, wait to add the sugar and broil until right before serving.