Beat butter, brown sugar, and white sugar together with an electric mixer in a large bowl until smooth. Add egg and vanilla extract and beat until smooth.
Mix flour, salt, and baking soda together in a separate bowl; add to butter mixture and blend until just incorporated into a dough.
Beat brownie mix, milk, water, and vegetable oil together in a separate bowl; add to dough and stir gently to keep the two batters distinct but somewhat evenly mixed.
Drop dough onto ungreased baking sheets by heaping tablespoonful.
Bake in preheated oven until set in the middle and crackling on top, 10 to 12 minutes. Cool cookies on sheets for 5 minutes before moving to a wire rack to cool completely.
