Step 1Heat the oven to 350 degrees.
Step 2Cut one-fourth cup butter into pieces and add it to a small saucepan or skillet. Heat over medium heat until the butter browns but does not burn. Pour the browned butter into a small bowl and refrigerate until the butter solidifies.
Step 3With the tip of a knife, scrape the seeds from the vanilla bean halves into the milk and then add the pods. Heat the milk until it's warm. Remove it from the heat and set aside to steep and cool.
Step 4Melt the remaining butter in a small, nonstick skillet. Remove it from the heat and add the brown sugar, stirring until the sugar dissolves. Pour the mixture into an 8-by-2-inch buttered cake pan, evenly coating the bottom of the pan.
Step 5Cut the plums in half, then cut each half into 3 or 4 slices. Arrange the slices in a single layer of concentric circles, beginning at the outside edge of the pan, so they are touching but not overlapping. Cover the bottom of the pan, then arrange a second layer of fruit on top. Set aside while preparing the cake batter.
Step 6Beat the cooled browned butter until light and fluffy. Beat in the sugar until creamy. Beat in the egg until blended.
Step 7In another bowl, stir together the cake flour, salt and baking powder. Remove the vanilla bean from the milk.
Step 8Fold the flour mixture into the butter-sugar mixture alternately with the vanilla-bean milk. Begin and end with the flour mixture.
Step 9Gently spoon the batter over the top of the sliced plums in the cake pan. Spread the batter over the fruit so all the fruit is covered and the batter is distributed to the sides of the pan.
Step 10Bake until the cake tests done in the center and is golden brown, about 50 to 55 minutes.
Step 11Remove the cake from the oven and let it stand for a few minutes. Run a knife around the sides of the cake to loosen the edges. Put a serving plate on top of the cake and invert. Carefully remove the baking pan from the cake. Cool. Serve warm.