For the custard: Bring the heavy cream and vanilla to a boil in a medium saucepan over medium-high heat. Remove from the heat and let cool for 10 minutes.
Whisk together the brown and granulated sugars and yolks in a large bowl until the mixture is combined and creamy.
Slowly drizzle the cooled cream mixture into the sugar mixture and whisk until it is combined and frothy. Set aside.
For the bananas foster filling: Combine the brown sugar, butter, and rum in a medium skillet. Cook over medium-high heat, stirring constantly, until the butter is completely melted. Use a wooden spoon to combine the bananas with the sauce, then cook until it thickens and is a nice creamy consistency, about 1 minute. Set aside.
Preheat the oven to 325°F. Heat a kettle of hot water to boiling.
Add five 8-ounce ramekins to a large roasting pan or baking dish large enough to fit them all without touching. Carefully add enough boiling water to the pan so that it goes about halfway up the sides of the ramekins.
Divide the filling (about 1/4 cup each) and custard (about 1/2 cup each) evenly among the ramekins. Bake until the custard is set and only jiggles slightly, 40 to 45 minutes.
Carefully remove the ramekins from the hot water bath and let them cool completely. Cover and allow to chill overnight in the refrigerator.
Once you are ready to serve, heat the rum in a small saucepan. Add the banana, butter, and sugar to a medium skillet. Once the butter is melted, pour the rum over the banana mixture and light with a lighter (this step is completely optional!). Stir and swirl so that the flame goes out. Add a few banana slices onto each ramekin, leaving any extra sauce behind. Sprinkle each ramekin with a teaspoon of granulated sugar. Then use a kitchen torch to heat the sugar until it is melted and hard. Garnish each with a few berries and a sprig or a few leaves of mint.
Use extreme caution when igniting alcohol. Remove the pan from the heat source before adding the alcohol. Pour the alcohol into the pan and carefully ignite with a match or click lighter. Return the pan to the heat and gently swirl to reduce the flames.
Propane gas torches are highly flammable and should be kept away from heat, open flame and prolonged exposure to sunlight. They should be used only in well-ventilated areas. Follow torch manufacturer's instructions for use.