Preheat oven to 350 degrees F and grease 2 8-inch round baking pans, line the bottom of each cake pan with parchment paper to form a circle to fit the bottoms. Set aside.
In a medium mixing bowl, whisk together the flour, baking powder, and baking soda.
In a large mixing bowl with an electric hand mixer or in the bowl of a stand mixer with a paddle attachment, cream together the butter and sugar until light, fluffy, and pale for a few minutes.
Add in the oil and vanilla bean paste, mix until fully combined, smooth and mixture is creamy. Add in the whole egg and egg yolk one at a time until incorporated.
Rotate between adding the buttermilk and flour mixture in 3 increments. Make sure the cake batter is combined, but do not over-mix.
Pour the cake batter into the prepared cake pans. Bake for 40-45 minutes or until a tooth pick inserted comes out clean and the tops are a light golden brown. Let the cakes cool on a cooling rack for about 30 minutes. Carefully run a butter knife along the edge of the cake pan and carefully flip it over on a cooling rack to cool for another 10-15 minutes or until cooled completely.