Start with the ice cream base. In a medium saucepan, mix together heavy cream, milk, 1/4 cup brown sugar, and salt. Heat the cream mixture over medium-high heat. When bubbles form around the edge of the pan, take off the heat.
In a medium bowl, whisk egg yolks and remaining brown sugar until smooth.
Slowly pour the hot cream (about 1/4 cup at a time) into the yolk mixture, whisking constantly. Add about 1/2 cup more cream into the yolks.
Pour the yolk mixture into the cream. Put back on the stove and cook over medium heat until the mixture thickens enough to coat the back of a spoon.
Strain the mixture through a fine-mesh strainer. Add vanilla. Let cool in a water bath. Cover with plastic wrap and refrigerate 2 hours or overnight.
While the ice cream base is cooling, start the cookie dough bits. Cream together butter, both sugars, and salt.
Add vanilla and cream. Mix in flour and stir until combined. Add chocolate chips. Cover and chill for at least 30 minutes.
Line a baking sheet with parchment, and drop cookie dough (about 1/2 teaspoon) onto sheet. Freeze until ready to use. This is a perfect job for little kitchen helpers. And they only ate half of the dough while they were making the bits. :)
Freeze the ice cream base according to manufacture’s instructions. When the ice cream is the consistency of soft serve, transfer it to a freezer-safe container.
Stir in the frozen cookie dough bits and freeze until firm.