Preheat oven to 300 degrees and have 6 1/2-cup ramekins or other oven-safe dishes ready.
In a large bowl, whisk together the egg yolks and 3 tablespoons of sugar and set aside.
Bring the cream and vanilla to a boil, then reduce the heat and simmer for a few minutes.
Very slowly, pour the hot cream into the egg yolk mixture and whisk vigorously so the eggs don’t scramble. Strain the custard and pour into the ramekins.
Place the ramekins into a baking dish and pour enough hot water into the dish to reach about 1/2-inch below the rims.
Bake for about 40-45 minutes, until the custard is set.
Let them cool and refrigerate overnight.
About 20 minutes before serving, slice bananas in long crosswise strips to fit in ramekins. Combine with rum, sugar and salt and let stand. Before serving, arrange bananas on top of cold custard. Use a strainer to sig brow sugar evenly over bananas. Use a torch to carefully melt sugar evenly over tops of custard. Or set under broiler, watching carefully so it doesn’t burn (a few burned spots are OK). Serve immediately.