01 - Preheat oven to 350°F. Line a 9x13-inch baking pan with parchment paper, leaving overhang on sides for easy removal. Alternatively, grease the pan lightly with butter.
02 - Combine the yellow cake mix, melted butter, and egg in a mixing bowl. Mix with a wooden spoon or hand mixer on low speed until a soft, cohesive dough forms. Press the mixture evenly into the bottom of the prepared pan using your hands or the back of a spoon.
03 - Beat the softened cream cheese in a separate bowl until completely smooth, about 2 minutes. Add eggs one at a time, mixing well after each addition. Gradually incorporate the powdered sugar, followed by lemon juice, lemon zest, and vanilla extract. Continue beating until the mixture is fluffy and smooth, about 1-2 minutes.
04 - Pour the lemon cream cheese mixture over the prepared crust, spreading evenly with a spatula. Bake for 35-40 minutes. The edges should be set and lightly golden, while the center remains slightly jiggly—this is essential for achieving the characteristic gooey texture.
05 - Allow the cake to cool completely in the pan, about 1-2 hours. The center will set further as it cools. Lift from the pan using parchment paper overhang and cut into 12 squares. Dust generously with additional powdered sugar before serving, if desired.