Mix brown sugar and butter together in a large bowl until light and fluffy, about 3 minutes. Add vanilla extract and beat until combined. Add flour in batches, alternating with condensed milk, mixing dough until combined. Stir in 2 cups chocolate chips.
Line a 9x11-inch baking pan with parchment paper. Press the cookie dough into the pan using a spatula. Cover with plastic wrap and refrigerate at least 3 hours.
Combine peanut butter and 2/3 cup chocolate chips in a microwave-safe bowl; heat in the microwave until smooth, about 45 seconds. Pour over the cookie dough and spread evenly. Chill for 1 hour.