Preheat the oven to 350 degrees.

Grease a 9-by-9 inch square pan with the neutral oil.

Place the butter, sugar, and lemon zest in the bowl of a stand mixer fitted with the paddle attachment. On medium speed, cream the mixture for about 4 to 6 minutes, until it's fluffy and smooth. Add the four eggs into the mixture, one by one, mixing each egg into the batter before adding the next. Add the flour and baking powder and mix on low speed just until the flour is incorporated, about 3 minutes.

Pour the batter into the greased pan. Evenly spread the batter with a spatula. 

Gently place the apricots halves, facing down (peel side up) onto the batter. 

Transfer to the oven and bake for about 30 to 35 minutes, until a toothpick inserted into the middle of the cake comes out clean.

Let the cake cool and serve.