Place milk, 1 cup of cream, sugar, and salt in a medium sauce pan. Heat until the mixture is steaming, but not boiling. Remove from heat.
Add yolks to a medium bowl and very slowly whisk in warm milk. Transfer mixture back to saucepan and set over medium heat. Stir constantly until mixture thickens slightly and is 170 degrees. Remove from heat immediately so the mixture doesn't scramble.
Place remaining heavy cream in a large bowl and place strainer over bowl. Pour thickened yolk mixture through strainer to remove lumps. Stir with cream. Add vanilla extract, 3 whole Oreo cookies, and 3 whole chocolate chip cookies. Stir together. Cover with plastic wrap.
Chill mixture until completely cold, about 4-6 hours. Once chilled, ice cream base is ready for your machine. Cookies should break down in the milk and resemble crumbs, but if they don't, use a spoon to break them up. Freeze ice cream according to machine instructions.
Once mixture is frozen, layer ice cream with chopped Oreos and chocolate chip cookies. I did two layers to make sure every bite had tons to cookies. Freeze ice cream until solid, about 8 hours.