Preheat the oven to 350ºF (180ºC). Grease and flour two 9-inch round cake pans and line the bottoms with parchment paper.
In a medium-sized bowl, whisk together the cake flour, baking powder and salt until well blended.
In a large bowl, beat the butter, sugar, oil and vanilla bean paste with an electric mixer at medium speed, until pale and fluffy, about 3 minutes.
On medium speed, beat in the egg whites in two additions, then beat in the egg until incorporated. On low speed, beat in one third of the flour mixture, then beat in the sour cream just until combined. Beat in the remaining flour mixture in two additions, alternating with milk. Beat just until combined.
Transfer the batter to prepared cake pans. Bake until the tops of the cakes are light golden brown and a toothpick inserted in center comes out clean, about 28-33 minutes. Do not over bake.
Allow the cakes to cool in the pans for 10 minutes, then remove from the pans and transfer to a cooling rack.
Cut each cake into four 2 3/4-inch rounds using a cookie or biscuit cutter.
Using a pastry brush, brush the mini cakes generously with lilac syrup or vanilla simple syrup.