Preheat oven to 350 degrees F. Line and grease an 8 or 9-inch springform or baking pan.
In the bowl of a stand mixer or using a hand mixer, cream butter and sugar until light and fluffy.
Add eggs, and lemon juice and mix.
Sift flour, baking powder, salt, cardamom and ginger in a medium bowl. Add flour mixture to butter and eggs, mixing just until well blended, scraping down the sides of the bowl.
Rub the lemon zest and sanding sugar together until mixed.
Spread the batter into the prepared pan. Push plum halves into the batter. Sprinkle the top of the cake with lemon sugar.
Bake in center of the oven at 350 degrees F. for 50-60 minutes or until top is golden brown and tester inserted in center releases with soft moist crumbs.
Allow cake to cool for 15 minutes, then remove springform and let cake finish cooling on a wire rack. Serve warm or room temperature.
Plum cake will stay fresh and moist well wrapped on the counter. No need to refrigerate!