Preheat oven to 400 degrees. Butter two 9-inch round cake pans. Whisk together flour, baking powder, and salt. In a separate bowl, combine 3/4 cup sugar, the milk, oil, and egg. Fold into flour mixture.
Divide batter evenly between prepared pans, and smooth tops. Arrange plums, cut sides up, over batter.
Combine cinnamon and remaining sugar, and sprinkle over plums. Dot with butter. Bake until tops are dark golden, plums are soft, and a toothpick inserted into center of cakes comes out clean, 30 to 35 minutes. Let cool. (Cakes are best the day you make them; leftover cake is great for breakfast the next day.)
To ensure the fruit retains its shape during baking, use plums that are almost — but not quite — ripe. They shouldn’t be rock hard; rather, they should be soft yet slightly firm. You can use apricots, nectarines, or peaches in place of the plums.