Center a rack in the oven and preheat the oven to 350? F.
Have at hand a 9x12 inch baking pan, preferably Pyrex or porcelain.
Cut each plum into 4 to 6 pieces and toss into a bowl. Add 1/2 cup of the sugar and all of the cinnamon and ginger. Stir the plums around and then let them sit, stirring from time to time, while you prepare the batter. They will give up some juice and a syrup will develop.
Put the remaining 1 cup sugar, the flour, baking powder and salt in another bowl. Whisk to blend, then pour in the milk and extracts and whisk again, so you have a nice smooth batter.
Put the butter in the baking pan and melt it in the oven, about 5 minutes. Remove the pan from the oven.
Give the batter a light whisking and pour it over the hot butter. The batter will probably set around the edges immediately. Scatter the plums over the batter and drizzle over whatever syrup has accumulated in the bowl.
Bake for 50-55 minutes, or until the top is golden and a thin knife inserted into the center of the cake comes out clean. Transfer the cake to a rack and cool in the pan for at least 20 minutes, or let cool to room temperature.