Preheat the oven to 350°F.
Grease a 9x13-inch baking pan thoroughly with nonstick cooking spray or butter.
Place the eggs in a large mixing bowl.
Beat the eggs with an electric mixer until evenly blended.
Add the yellow cake mix, instant lemon pudding mix, water, and oil to the beaten eggs.
Mix on medium speed until the batter is smooth and well combined.
Scrape down the sides and bottom of the bowl as needed to ensure no dry cake mix remains.
Pour the batter into the prepared baking pan.
Spread the batter evenly in the pan using a spatula.
Place the pan in the preheated oven.
Bake for 30 to 35 minutes, or until a toothpick inserted into the center of the cake comes out clean.
While the cake is baking, prepare the soaking syrup.
In a medium bowl, combine the melted unsalted butter, lemon juice, water, and powdered sugar.
Stir until the syrup is smooth and pourable.
Remove the cake from the oven once fully baked.
While the cake is still warm, poke holes evenly over the surface using a skewer or fork.
Slowly pour the lemon soaking syrup over the warm cake, distributing it evenly across the surface.
Allow the syrup to absorb into the cake.
Let the cake cool completely in the pan before slicing.
Serve once fully cooled.