Cook the Apples: Combine the brown sugar and butter in a medium skillet over medium heat. When the butter is melted and bubbly, add the apples and cinnamon. Cook, stirring frequently, until the apples are softened and most of the liquid has evaporated, 10 to 15 minutes. Remove from the heat and let cool. Store in an airtight container in the refrigerator until ready to use.
Make the Ice Cream Base: In a medium bowl, whisk together the egg yolks, then whisk in 2½ tablespoons of the sugar. Set aside.
In a medium saucepan, stir together the cream, milk, salt and remaining sugar over medium-high heat. When the mixture approaches a simmer, reduce the heat to medium.
Ladle some of the hot cream mixture into the egg yolk mixture, whisking constantly. Repeat once more. Using a rubber spatula, stir the cream in the saucepan as you slowly pour the egg-and-cream mixture from the bowl into the pan.
Cook the mixture over medium heat, stirring constantly, until it is thickened, coats the back of a spatula and reaches 170 to 175 degrees F on an instant-read thermometer. Strain the base through a fine-mesh sieve into a clean bowl. Set the bowl in an ice water bath and stir occasionally until cool. Cover and refrigerator overnight.
Churn the Ice Cream: When you're ready to puree the ice cream, combine the cooked apples with the vanilla extract and half of the chilled custard base in a blender or food processor. Puree until smooth.
Combine the pureed mixture and the remaining custard base in the bowl of your ice cream machine and freeze according to the manufacturer's instructions. Add the chopped cookies during the last minute or so of churning, or fold them by hand when it has finished. Transfer the ice cream to a freezer-safe container and freeze.