Preheat the oven to 350 degrees Fahrenheit. Lightly grease the bottom and sides of a 9" square baking pan with cooking spray or oil and line with parchment paper. Set aside.
In a mixing bowl, whisk the eggs, vanilla, salt, and maple syrup. Add in the baking soda and half the almond flour. Stir well to combine. Stir in the butter and remaining flour.
Transfer the batter to the prepared baking pan and spread into an even layer. Bake 15-18 minutes or until set and an inserted toothpick comes out clean. Let the cake cool completely.
For the "egg" topping
Drain the apricots, collecting the juice. Place the apricot halves, cut side down, on a paper towel to dry.
In a bowl, whisk the pudding mix with 1 cup milk until it thickens. Add sour cream and mix until combined. Spread the "egg white" evenly over the cooled cake using an offset spatula.
Distribute apricot halves on top of the pudding and glaze with reserved juices using a pastry brush. Refrigerate before serving.
You can easily double this recipe if you're serving it to more than 9 people. Either use two 9" square pans or one 10x15-inch pan.
Lining the pan with parchment paper ensures easy removal of the cake. You can either serve it from the pan or place the cake base on a serving plate first and then spread the topping. Either way, you don't want it to stick to the bottom of the pan.
I highly recommend Del Monte apricots (not an affiliate). The halves are firm and perfectly shaped. Some other brands I tried were less than desirable - they were too soft and fell apart.
You can use vanilla pudding if you wish, but most have a yellow hue which is not suitable for "egg whites". Even the white chocolate pudding was off white, but the addition of sour cream turned it white enough.
A sprinkle of fake salt and pepper will make these eggs even more convincing. I used white crystal sprinkles for the salt, grated dark chocolate as pepper, and green candy shreds for chives. You can omit the "salt" and also use cocoa powder or ground vanilla bean powder as your pepper. Chopped pistachios make good chives as well.