Preheat the oven to 350°F. Prepare a 10x10-inch square pan or a 13x9-inch pan by greasing it with vegetable shortening and lining with parchment paper that overhangs all four of the pan’s edges.
Mix all of the dry ingredients on slow speed to blend. Add the soft butter and mix until evenly crumbly, like fine damp sand. It may form a paste, depending on the temperature of the butter, how much it's mixed, and granulation of the sugar used.
Add the egg whites one at a time, then the whole egg, beating well after each addition to begin building the structure of the cake. Scrape down the sides and bottom of the bowl after each addition.
In a small bowl, whisk the yogurt (or milk) with the vanilla and lemon extracts. Add this mixture, 1/3 at a time, to the batter. Beat 1 to 2 minutes after each addition, until fluffy. Be sure to scrape down the sides and bottom of the bowl.
Pour the batter into the prepared pans. Bake for 23 to 26 minutes, or until a toothpick tester comes out clean. Let the cake cool completely in the pan.
Vanilla pudding and apricot topping
Pour the juice from the canned apricot halves and place them on paper towels to drain while you prepare the pudding.
Place 3/4 cup of the whole milk in a bowl. Stir in the sugar, vanilla and cornstarch until smooth and clump-free.
Place the remaining milk in a large pot and bring to a boil over medium-high heat. When the milk bubbles, remove the pan from the heat and whisk in the cornstarch mixture. Place the pan back on the heat source and continue to cook and whisk constantly until the mixture thickens.
Pour the pudding over the cooled cake. Place the apricots, evenly spaced, cut-side-down on the cake (I used 9 apricot halves on my 10x10 cake, but you may need a dozen for a 9x13 cake).
Place the cake in the refrigerator until the pudding is set.
Whisk together the powdered gelatin and water in a small microwave-safe bowl. Let the mixture stand until solid. Place the mixture in the microwave and heat for 10 seconds, or until the gelatin is liquefied. Stir in the caster sugar until well incorporated.
Use a pastry brush to distribute the gelatin glaze over the surface of the cake and apricots. This will give it a shiny appearance, similar to that of a fried egg.
Cover the cake with plastic wrap and refrigerate until the glaze is set. When ready to serve, run a knife between the cake and the pan, and then lift the cake out of the pan with the overhanging parchment. Cut the cake into squares, each with an apricot in the center. Serve chilled or at room temperature. Refrigerate leftovers.