Combine the butter and both sugars in a microwave-safe bowl. Heat in 15-second intervals, stirring after each one until the butter is fully melted. Whisk until smooth.
Whisk in the milk, vanilla, and salt.
Add the flour and use a rubber spatula to fold it in until fully incorporated.
Fold in the chocolate chips.
Cover with plastic wrap and freeze for 15-20 minutes to firm up.
Make the ice cream
Add the heavy cream to a large bowl. Using an electric mixer set to medium speed, beat until stiff peaks form.
Turn the speed to low and add the vanilla. Slowly pour in the condensed milk. Turn the speed back up to medium and beat until thickened.
Using a silicone spatula, gently fold in the chocolate chips. Drop small pieces of the cookie dough into the ice cream and gently fold it in. (Aim for smaller pieces so you get more throughout the ice cream.)
Transfer to a freezer-safe container and freeze for at least 6 hours.
Notes
Heat treat the flour before you begin. Spread the flour on a baking sheet. Toast it for 10 minutes in a 350°F oven. Let the flour cool completely then run it through a sifter to remove any clumps that may have formed.