Preheat the oven to 350°F (175°C). Line a 9x9-inch pan with foil and spray it with cooking spray; set aside.
In a stand mixer fitted with the paddle attachment, combine the red velvet cake mix, softened butter, and egg. Mix until a thick dough forms.
Press about two-thirds of the dough evenly into the bottom of the prepared pan to form the base.
Sprinkle the white chocolate chips evenly over the pressed dough.
Warm the cream cheese if needed until very soft (use 20-second microwave intervals), then whisk together the cream cheese and sweetened condensed milk until smooth.
Pour the cream cheese and sweetened condensed milk mixture over the white chocolate chips in the pan.
Divide the remaining dough into small pieces, flatten them between your fingers, and space them evenly over the top of the bars; press lightly to flatten but it need not cover completely.
Bake for about 28–30 minutes, or until the top just begins to brown.
Allow the bars to cool completely in the pan before slicing so the filling sets and they are easier to cut.