In a small saucepan over low heat, combine the water and cinnamon stick and simmer until fragrant, about 3 minutes. Add the brown sugar and stir until it is dissolved. Remove from heat and let cool to room temperature.

Unwrap the frozen bananas, chop them coarsely, and set them aside.

Remove the cinnamon stick from the syrup and pour it into a blender or food processor. Add the cream and bananas and blend until smooth and mushy.

Pour into an ice cream machine freeze, following the manufacturer's instructions. Add the pecans and white chocolate to the ice cream a few minutes before the cycle is finished. Since the mixture is partially frozen, it won't take long to freeze, about 10 minutes. Serve with Brown Sugar Caramel Sauce (recipe below).

In a medium saucepan with high sides, stir the brown sugar, 1/2 cup water, and the corn syrup together. Stir gently so you don't splash any of the mixture onto the sides of the pan. Cook over medium-high heat, stirring until the sugar dissolves. Increase the heat to high and stop stirring. The mixture will begin to boil. Boil for 10 minutes. Remove from heat and slowly add the cream, whisking gently. The caramel will bubble vigorously when the cream is added, then will clump.

Return the pan to low heat and warm the mixture to dissolve the caramel clumps. When the caramel is smooth, add the vanilla and salt, if using, and whisk again.

If using the sauce immediately, remove from the heat and beat vigorously to cool it slightly. Let stand for a few minutes before using.

Let any leftover caramel come to room temperature, cover, and refrigerate. You can rewarm the sauce in the microwave. Will keep in the refrigerator for up to 5 days.