Combine half-and-half, brown sugar, white sugar, and salt in a saucepan over medium-low heat; cook and stir until sugars are dissolved, about 5 minutes. Remove saucepan from heat and pour 1/2 the mixture into a large bowl; whisk eggs into mixture until well combined.
Transfer egg mixture back into saucepan with remaining half-and-half mixture; stir in cream. Cook and stir mixture over medium-low heat until cream mixture is thickened and coats the back of a spoon, 5 to 10 minutes. Remove saucepan from heat and stir in vanilla extract.
Pour cream mixture over a fine mesh sieve into a bowl; cool to room temperature. Stir mashed bananas into cooled cream mixture.
Pour banana cream mixture into an ice maker and freeze according to manufacturer’s instructions. Fold vanilla wafers into ice cream during the last 5 to 10 minutes of freezing.