In a saucepan, combine whole milk, heavy cream, vanilla bean, salt, and brown sugar. Heat over medium heat until steaming.
In a mixing bowl, whisk together the egg yolks. Gradually add the warm milk mixture into the yolks, stirring constantly.
Return the mixture to the saucepan and cook, stirring, until it thickens. Remove from heat and cool.
For the cookie dough, cream together the butter, granulated sugar, and brown sugar. Mix in the salt, baking soda, vanilla, and milk.
Gradually mix in the flour until combined, then fold in the chocolate chips.
Chill the ice cream base in the fridge, then churn in an ice cream maker according to the manufacturer's instructions.
Add in chunks of cookie dough while churning, then transfer to a container and freeze until firm.