Preheat the oven to 350 degrees Fahrenheit. Grease a deep 9 x 13-inch baking pan and line it completely with parchment paper. Set is aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat together the egg yolks and sugar for 1 minute at medium low speed. Then add the oil, yogurt, vanilla extract, and salt, mixing well to emulsify.
Transfer the yolk mixture to a separate large bowl and set aside. Thoroughly clean the stand mixer bowl (you will be using it to beat egg whites, and if there's a speck of oil on the bowl the whites won't beat properly.
In the bowl of a stand mixer fitted with a whisk attachment, beat the egg whites until they form soft peaks. Set aside.
Add the flour and baking powder to the yolk mixture and stir to combine. Gently fold the egg whites into the batter until incorporated.
Pour the cake batter into the prepared pan, then arrange the apricot halves on top, cut-side up. Bake for 25 to 35 minutes, or until the cake is golden on top and firm to the touch. Remove from the oven and allow to cool in the pan, then dust with the powdered sugar and drizzle with the honey (optional) and cut into pieces to serve.