Position a rack in the center of the oven and preheat the oven to 375 degrees. Line two baking sheets with parchment paper.
In a medium bowl, stir together both flours, baking powder, and salt.
In the work bowl of a stand mixer fitted with the paddle attachment, beat together the butter, shortening, and brown sugar until light and creamy, about 3 minutes. Add the eggs and the buttermilk and beat until combined.
In a measuring cup, combine the milk, baking soda, and vinegar. Add the milk mixture to the batter along with the flour mixture and beat on low speed until just combined. Add the vanilla and beat on medium speed for about 2 minutes, until completely combined.
Using a spoon, drop about 1 tablespoon of batter onto one of the prepared baking sheets and repeat, spacing them at least 2 inches apart. Bake one sheet at a time for about 10 minutes each, or until the cakes begin to brown. Remove from the oven and let the cakes cool on the sheet for at least 5 minutes before transferring them to a rack to cool completely.
Spread a layer of marshamallow filling and then a layer of chocolate ganache on the flat side on one cake and then top with another cake, flat-side down.