In a bowl, whisk together milk, brown sugar, vanilla, and salt until sugar is fully dissolved.
In a separate chilled bowl, whip heavy cream until stiff peaks form.
Gently fold milk mixture into whipped cream using a spatula.
Fold in cookie dough pieces and chocolate chips.
Transfer to a container, cover tightly (press wrap directly onto the surface to prevent ice crystals), and freeze for at least 6 hours.