To make the fruit filling, melt the butter in a 12-inch heavyweight saute pan over medium-high heat. (A large saute pan is critical because as the plums cook they need to release steam.) Sprinkle the sugar over the melted butter and then carefully distribute the plum wedges evenly across the pan. Toss in the vanilla bean, followed by the salt. Allow the plums to cook, undisturbed, until they begin to release their rosy juices. Try to resist stirring the pan for at least 3 to 4 minutes to encourage a slight caramelizing of the flesh of the fruit. Once the juices begin to come to a steady simmer, turn the heat down to medium and cook the fruit down (you can stir now, and do so frequently!) to the texture of a thick applesauce, about 25 minutes. It is important to cook the fruit long enough to make a thick filling in order to stack a sturdy cake. Fish out the vanilla bean and put the filling, uncovered, in the refrigerator to chill for at least 4 hours, stirring it occasionally to help the cooling process. Your plum filling should yield a generous 2 cups. Made in advance, it can keep for 2 days refrigerated.
Center an oven rack and preheat the oven to 350 degrees F.
Sift together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, cloves, and salt in a bowl, then whisk the mixture to ensure that the ingredients are well mixed.
In the bowl of a stand mixer fitted with the paddle attachment, blend the brown sugar and the butter together on low speed until the butter has melded into the sugar and the mixture resembles wet sand. As you make the batter, stop the mixer frequently and scrape the paddle and the sides of the bowl with a rubber spatula. Still on low speed, drizzle the oil into the mixture. Turn the mixer up to medium-high speed and beat until the batter is fluffy, about 3 minutes. Blend in the eggs one at a time at low speed, adding the next one as soon as the previous one has disappeared into the batter. With the mixer still on low speed, add the flour mixture in three parts, alternating with the buttermilk in two parts, beginning and ending with the flour. After each addition, mix until just barely blended and stop and scrape the bowl. Stop the mixer before the last of the flour has been incorporated and complete the blending by hand with a rubber
Although you will bake only three cakes at a time, you will do this process twice for a total of six cakes. There is no need to refrigerate the batter while it waits its turn. Pour 1 cup (9 1/2 ounces by weight) of batter into each of the prepared pans and bake in the middle of the oven until the centers are just barely firm when lightly touched (the cake will be very blond but have a golden ring around its edge), approximately 15 minutes. Remove the cakes to a wire rack and promptly run a small, thin knife around the edges. Let the cakes cool for 10 minutes before removing them from their pans. Leave the parchment papers on until you assemble the cake. Wipe out the pans with a paper towel, then regrease and repaper them before baking the next three cakes. Once all the layers are baked, be sure to let them all cool to room temperature before assembling the stack.
To assemble the cake, place one of the layers top side up on a flat serving plate. Using a thin metal spatula, spread no more than 1/3 cup of the chilled fruit filling evenly over the layer. Align the next cake layer, again top side up, on top of the previous layer, and repeat with another thin layer of plum filling. Continue with the next four layers of cake and filling, leaving off the filling from the top layer. Whisk together the sifted confectioners' sugar, milk, and vanilla. Pour this glaze over the middle of the cake and allow it to drip down the sides of the stack. Any leftover filling can be refrigerated to spread on your morning toast.
This cake is best served at room temperature. Stored in an airtight container, it keeps for up to 5 days.
NotesUse three 8-inch round cake pans, greased and bottoms lined with parchment paper circles.