In a saucepan, combine the half and half, brown sugar, white sugar, and salt. Stir over medium-low heat until the sugar dissolves.
Remove the saucepan from heat. Place the beaten eggs in a bowl. Gradually whisk in about half of the sugar mixture to the eggs, going slowly to prevent the eggs from cooking.
Pour the egg mixture into the saucepan with the remaining half and half. Stir in the cream.
Cook the mixture over medium heat for 5-10 minutes, stirring frequently, until it thickens.
Remove the saucepan from heat and stir in the vanilla extract. Pour the mixture through a fine-meshed sieve into a bowl and let cool to room temperature.
After the mixture has cooled, stir in the mashed bananas and pour into an ice cream maker. Freeze according to the manufacturer's instructions. About 5 minutes before the end of the freeze time, add the Nilla wafers.
If the ice cream is too soft, transfer it to a container and set in the freezer for another 1-2 hours