Follow these steps carefully for perfectly gooey results:

1. Preheat and Prepare: Preheat your oven to 350°F (175°C). Grease a 9×13 inch baking pan thoroughly with butter or non-stick cooking spray. For easier removal and cleanup, line the pan with parchment paper, leaving an overhang on two sides to act as handles. Grease the parchment paper as well.
2. Make the Crust: In a large bowl, combine the dry yellow cake mix, ½ cup melted butter, and 1 lightly beaten egg. Stir together using a sturdy spoon or electric mixer on low speed until just combined. The mixture will be thick and somewhat crumbly.
3. Press Crust into Pan: Transfer the crust mixture to the prepared baking pan. Using your fingers, the back of a spoon, or the bottom of a measuring cup (lightly floured or greased if sticking), press the mixture evenly across the bottom of the pan to form a solid base layer. Ensure it reaches the edges.
4. Make the Filling: In a separate large bowl (or the same bowl after rinsing), place the softened cream cheese. Using an electric hand mixer or a stand mixer fitted with the paddle attachment, beat the cream cheese on medium speed until smooth and creamy, with no lumps remaining. This step is crucial for a silky filling.
5. Add Remaining Filling Ingredients: Add the 2 lightly beaten eggs and the vanilla extract to the cream cheese. Beat again on medium speed until well combined and smooth. Scrape down the sides of the bowl as needed.
6. Incorporate Powdered Sugar: Gradually add the sifted powdered sugar to the cream cheese mixture, beating on low speed initially to avoid a sugar cloud, then increasing to medium speed. Mix until the sugar is fully incorporated and the mixture is smooth and thick.
7. Mix in Butter: Pour in the remaining ½ cup of melted butter. Beat on low speed until the butter is fully incorporated and the filling is glossy and smooth. Be careful not to overmix once everything is combined.
8. Pour Filling Over Crust: Carefully pour the cream cheese filling mixture evenly over the pressed crust layer in the baking pan. Use a spatula to gently spread it out, ensuring it covers the crust completely and reaches all corners.
9. Bake: Place the pan in the preheated oven. Bake for 40-50 minutes. The edges should be golden brown and slightly puffed, and the center should be mostly set but still slightly jiggly or gooey. It will continue to set as it cools. Avoid overbaking, as this will result in a drier, less gooey texture. Start checking around the 40-minute mark.
10. Cool Completely: Remove the pan from the oven and place it on a wire rack. Let the gooey butter cake bars cool completely in the pan. This is a critical step – cutting them while warm will result in a messy, unset center. Cooling can take 2-3 hours at room temperature, or you can speed it up slightly by placing it in the refrigerator after about an hour of cooling at room temperature.
11. Cut and Serve: Once completely cooled, use the parchment paper overhangs to lift the entire slab out of the pan onto a cutting board. Dust generously with additional powdered sugar, if desired. Using a sharp knife (wiping it clean between cuts can help for neat squares), cut into bars or squares of your desired size (16-24 bars is typical).