Prepare Cookie Dough:

1. Using an electric mixer, cream butter and both sugars until light and fluffy, about 1 minute.
2. Beat in milk and vanilla extract until well combined.
3. Mix in flour and salt until just incorporated.
4. Fold in chocolate chips by hand.
5. Refrigerate mixture while preparing ice cream base.

Make Ice Cream Base:

1. In a small saucepan, melt butter over medium-low heat.
2. Continue cooking, stirring occasionally, until butter turns golden brown and develops a nutty aroma.
3. Transfer to a medium bowl and stir in brown sugar.
4. Allow to cool to room temperature, approximately 15 minutes.
5. Whisk in sweetened condensed milk and vanilla extract until smooth.
6. In a separate large bowl, whip heavy cream to stiff peaks using an electric mixer.
7. Gently fold half the condensed milk mixture into whipped cream.
8. Add remaining condensed milk mixture and fold until just combined.
9. Fold in most of the mini chocolate chips, reserving some for topping.

Assemble and Freeze:

1. Spread half the ice cream mixture in a freezer-safe container.
2. Break cookie dough into small chunks and distribute half over ice cream.
3. Add remaining ice cream mixture and top with remaining cookie dough pieces and reserved chocolate chips.
4. Cover with plastic wrap pressed directly against surface.
5. Freeze overnight or until completely firm.
6. Before serving, allow to soften in refrigerator for 15-20 minutes.