Prepare the cookie dough: Cream butter, sugar, and brown sugar together. Add milk and the vanilla, mix well. Mix in the flour (see notes below) and salt until everything is combined. Fold in the chocolate chips. Make small balls of dough the size of 1/2 teaspoon. Place in the fridge until ready to use.
Prepare the ice cream: Combine brown butter and heavy cream in a large sauce pan. Whisk until well combined. Heat to a simmer.
In a separate bowl (heatproof) mix together brown sugar, egg yolks, and salt.
Slowly pour in the warm cream mixture, whisking constantly. Pour back into the saucepan, and heat over medium-low heat. Stir constantly, until the mixture thickens to coat the back of a wooden spoon (170-175 degrees).
Pour the mixture through a strainer into a large bowl. Add whole milk and vanilla; stir until well blended.
Cover and place in the fridge until cold. Can take up to 8 hours, over night works great.
Pour the custard into your ice cream maker, and freeze according to the instructions.
Once frozen, transfer to an air tight container. Stir in pieces of cookie dough until evenly distributed. You may have more than you need, depends on how chunky you like it.
Cover and freeze about an hour before serving.
Notes
To make the cookie dough 100% safe to eat you want to bake the flour for 5 minutes at 350 degrees F. Cool completely before making the dough.