Grease a 10 inch tart pan with a removable bottom.
Beat the butter and sugar until very light in color, and very fluffy, about five minutes. Add the eggs, one at a time, and incorporate them in well. Add the zest of the lemons and vanilla extract.
Sift flour with baking powder, then add the flour mixture to the butter, sugar mixture, alternating with the milk. Mix well until it becomes a smooth, soft dough. Transfer the dough to the tart pan and smooth it with a spatula.
Place the cut plum sections, skin-side down, onto the dough.
Bake about 1 hour or until a toothpick comes out clean.
Cool the cake on a rack, then transfer it onto a serving platter. Dust with powdered sugar.
Serve with softly whipped heavy cream sweetened with powdered sugar and flavored with vanilla before serving.