2. Beat the butter in a mixing bowl with an electric mixer on high speed for 30 seconds. Gradually add the sugar and continue to beat until the mixture is pale yellow and very fluffy.
3. Add the eggs 1 at a time while mixing on low speed. Stir in the vanilla.
4. Sift the flour, baking powder and salt together.
5. Add half of the flour mixture and blend, then add the milk. Finally, add the rest of the flour mixture. The batter should be smooth. Do not overmix.
6. Coat a 9" round cake pan with nonstick spray. Pour the batter into the pan. Arrange the apricot halves on top in a single layer.
7. Bake until golden brown and set in the center, about 25 to 30 minutes.
For the Apricot Sauce:
1. Bring the water and sugar to a boil in a small saucepan over high heat. Remove from the heat.
2. Puree the apricots in a blender with half of the sugar syrup. Adjust the taste by adding a few drops of lemon juice and more sugar syrup until a pleasant balance of sweetness and tartness is reached.
3. Putting it together: Place 1 slice of cake on each plate. Drizzle the sauce around the cake. Garnish with a mint sprig.