Preheat oven to 350°F.
In a bowl, mash cane sugar and softened butter until combined.
Add sifted flour, baking powder, kosher salt, and eggs; beat well to form batter.
Transfer batter to a 9-inch springform pan.
Arrange plum halves skin side up on top of batter, pressing lightly to set.
Sprinkle with reserved cane sugar, fresh lemon juice to taste, and cinnamon.
Bake for 40-50 minutes until a cake tester inserted in the center comes out clean.
Cool before serving. Can be wrapped in foil and frozen in a Ziploc bag for later.