Place the bowl of your ice cream maker in the freezer the night before. Add cashew milk and cashews to a high-powered blender and blend until completely smooth. Add all the remaining ice cream base ingredients EXCEPT the mini chocolate chips and blend together until well combined - about 2 minutes.
Transfer to a shallow container, cover in plastic wrap, and place in the refrigerator to let the ice cream base chill all the way through (about 2 hours).
While the ice cream base is chilling, make your cookie dough. Stir together sugars and coconut oil together in a small bowl until evenly mixed. Add apple sauce and vanilla extract, stirring to combine. Once combined, sift in the flour and mix until an even dough has formed. Fold in the chocolate chips. Roll dough into a long, thin log and cut into ¼" pieces. Place pieces on a baking pan and freeze for at least 20 minutes.
When the ice cream base is chill, transfer it to the frozen base of your ice cream maker and churn the ice cream according to the manufacturer's instructions. Once it is thick and creamy and resembles soft serve, add ½ cup mini chocolate chips and frozen cookie dough chunks. Churn for 30 seconds more or until chocolate chips and cookie dough is evenly distributed.
Transfer it to a loaf pan, cover, and place in freezer. Allow ice cream to freeze all the way through, about another 2 hours.
To serve, remove from freezer and allow ice cream to thaw for 10 minutes before scooping.
Store in an airtight container in the freezer for up to 2 weeks.