Line an 8" square baking pan with foil, leaving a 2" overhang on all sides; lightly coat the foil with cooking spray.
In a small sauce pan, melt butter over medium heat. Continue to cook, swirling pan occasionally, until deep golden brown. Immediately pour butter into a bowl and let cool for 3 minutes.
In a medium bowl, whisk together the flour, baking soda and salt.
In the bowl of a stand mixer fitted with a paddle attachment, or in a bowl with a hand-held mixer, beat the brown butter and sugars at medium speed until well blended. Let cool for 3 minutes. Add the egg and vanilla extract and continue to beat until well blended.
Gradually add the dry ingredients and beat on low speed until just combined. Add the chocolate chips and stir to combine.
Press about half of the dough evenly into the bottom of the prepared baking pan. Spread the condensed milk in an even layer over the dough, leaving a ¼" border around the edges. Sprinkle the chopped chocolate over the condensed milk. Squeeze off chunks of the remaining dough and sprinkle over the chopped chocolate, pressing gently to adhere.
Transfer to the refrigerator to chill for at least 30 minutes.
Preheat oven to 350 degrees with rack in center position.
Bake cookie bars until the top is lightly golden, 30-35 minutes.
Remove pan from the oven and let cool completely. Using the foil overhang, lift bars from pan and transfer to a cutting board. Cut into 16 2" squares.