Arrange a rack in the middle of the oven and preheat the oven to 350℉.
Line the bottom and the 2 long sides of a 9x13 baking pan with a sheet of parchment paper, making sure it is long enough to hang over the sides and form a sling.
Melt the butter in a small saucepan over medium heat and then transfer to a medium bowl. (Or melt the butter in the microwave using 20-second intervals.)
Add the cake mix and 2 of the eggs to the melted butter.
Beat with an electric hand mixer or in a stand mixer with the paddle attachment on medium speed until smooth, about 30 seconds.
Transfer to the prepared baking pan and spread into an even layer.
Wipe out the bowl and paddle (no need to wash). Add the cream cheese, remaining eggs, powdered sugar, and vanilla extract.
Beat on medium-low speed until smooth, scraping down the sides if necessary.
Pour over the top of the cake batter and spread into an even layer.
Bake until golden brown around the edges but the center is slightly jiggly and the pan is shaken, about 35-40 minutes.
Place on a wire rack and let it cool completely for 2-3 hours.
Run a thin knife around the edges to loosen. Grasping the paper, lift the slab out and onto a cutting board. Cut into 24 pieces.
Notes
Store the bars in an airtight container in the refrigerator for up to a week. To freeze, wrap in plastic and place in a freezer-safe container for up to 4 months. Thaw in the refrigerator for an hour or at room temperature for 20 minutes.