Preheat the oven to 350°F (175°C). Line a 9×9-inch metal baking pan with aluminum foil and lightly grease the foil.
In a medium bowl, combine 1 (approx 15 oz) box lemon cake mix, ½ cup (113 g) softened unsalted butter, and 1 large egg. Beat with a hand mixer or a sturdy spoon until the mixture comes together into a thick, cookie-dough–like batter.
Reserve about ½ to ⅔ of the dough for the topping. Press the remaining dough evenly into the bottom of the prepared pan to form a compact base.
Evenly drizzle ½ cup sweetened condensed milk over the dough layer, leaving a small border at the edges.
Sprinkle 1 cup (170 g) white chocolate chips evenly over the sweetened condensed milk.
Break the reserved dough into small pieces and crumble them over the top to mostly cover the chips. Press small pieces flat between your fingers and place them on top, sealing the edges first so the condensed milk does not bubble out.
Bake on the center oven rack for 28–33 minutes, or until the top is slightly browned and set.
Remove from the oven and let the pan cool completely on a wire rack. Use the foil to lift the bars from the pan, then cut into squares and serve.