How to Make It
Heat oven to 350° F. Butter an 8-inch cake pan and line the bottom with parchment paper.
Melt 1 tablespoon of the butter in a large skillet over medium-high heat. Add the plums and ¼ cup of the sugar and cook, tossing, until the sugar dissolves and the juices from the plums become syrupy, 3 to 4 minutes. Arrange the plums in the cake pan in slightly overlapping concentric circles, starting from the outside. Spoon any pan juices over the top.
In a small bowl, whisk together the flour, baking powder, baking soda, and salt.
With an electric mixer, beat the remaining ½ cup of the butter and ⅔ cup of the sugar until fluffy. Beat in the egg, sour cream, and vanilla. Gradually add the flour mixture, mixing just until incorporated.
Pour the batter over the plums and bake until a toothpick inserted in the center comes out clean, 50 to 55 minutes. Let cool in the pan for 1 hour. Place a large plate over the cake pan and invert the cake onto the plate.