Brown butter: Cut butter into pieces. Place in light-colored saucepan over medium heat. Stir frequently as it melts, foams, then subsides. When small brown specks appear and butter smells nutty and turns amber, remove from heat immediately. Pour into heatproof bowl, scraping all brown bits. Cool 10-15 minutes.
Make dough: In mixing bowl, combine cooled brown butter, brown sugar, and granulated sugar. Whisk until glossy. Add egg and vanilla, whisk until smooth and lightened. In separate bowl, whisk flour, baking soda, and salt. Fold dry ingredients into wet just until no flour streaks remain. Fold in mini chocolate chips. Do not overmix.
Shape and chill: Place dough between two sheets of parchment. Roll into 6.5x8-inch rectangle, about ¼-inch thick. Press extra mini chips gently into top. Transfer dough (still between parchment) to baking sheet. Refrigerate 2 hours or freeze 1 hour until firm.
Preheat and prep: When ready to bake, preheat oven to 175°C (350°F). Line baking sheet with parchment.
Cut and bake: Remove chilled dough. Cut into 10-12 sticks, about 1 inch wide. Place on prepared sheet, 1 inch apart. Bake 14-16 minutes until edges slightly golden. Cool on sheet 10 minutes, then transfer to wire rack.
Make dip: In microwave-safe bowl, combine chocolate chunks, 2 tbsp milk, and butter. Microwave in 15-20 second intervals, stirring between, until melted and smooth. Add remaining milk if thinner consistency desired.
Serve: Arrange cookie fries on platter with dip. Serve warm or at room temperature.
