Place brown sugar and butter in a bowl; beat with an electric mixer until creamy, 4 to 5 minutes. Mix in milk, vanilla, and salt. Add flour; mix thoroughly by hand. Fold 1 cup chocolate chips into the dough. Cover and refrigerate for about 1 hour.
Scoop out tablespoonfuls of cookie dough and roll into round balls. Arrange them on a plate or baking sheet that will fit in the freezer. Freeze until hard, about 1 hour.
Place 1 1/2 cup chocolate chips and white chocolate chips in separate microwave-safe bowls. Microwave until soft, about 30 seconds. Stir. Continue to microwave in 10-second intervals, stirring after each interval, until completely melted, 30 seconds to 1 minute.
Dip cookie dough balls in melted chocolate. Drizzle white chocolate on top. Refrigerate until firm, about 30 minutes.