In a medium bowl, whisk together the heavy cream, whole milk, granulated sugar, and fine sea salt until the sugar is completely dissolved.
Stir in the vanilla extract and add blue gel food coloring one drop at a time, stirring after each addition, until you achieve a deep, vibrant blue color.
Pour the blue ice cream base into your ice cream maker and churn according to the manufacturer's instructions until it reaches a thick, soft-serve consistency, which should take about 20-25 minutes.
Use a spatula to gently fold the coarsely crushed chocolate chip cookies and the mini chocolate chip cookies into the churned ice cream until they are evenly distributed throughout.
Transfer the ice cream to a lidded, freezer-safe container and press a piece of parchment paper directly onto the surface. Seal with the lid and freeze for at least 4 hours, or until firm and scoopable.