In a mixing bowl, whisk together the heavy cream, whole milk, and granulated sugar until the sugar is dissolved.
Add the lemon zest, lemon juice, and vanilla extract to the cream mixture. Stir until well combined.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions until it reaches a soft-serve consistency.
Once the ice cream has finished churning, gently fold in the crushed graham crackers or shortbread cookies until evenly distributed.
Transfer the ice cream to a freezer-safe container and smooth the top with a spatula.
Cover the container with plastic wrap or a lid and freeze the ice cream for at least 4-6 hours, or until firm.
When ready to serve, scoop the Lemon Icebox Pie Ice Cream into bowls or cones and enjoy!