Preheat the oven to 375°F (190°C). Grease a 9-inch round cake pan and dust it with flour, tapping out any excess.
In a medium bowl, whisk together the flour, baking soda, and salt.
In a large bowl, use an electric mixer on medium-high speed to beat the softened butter, granulated sugar, and brown sugar until pale and fluffy, about 3-4 minutes.
Add the eggs one at a time, beating well after each one. Mix in the vanilla extract.
Reduce the mixer speed to low. Add half of the flour mixture and mix until just combined. Add the sour cream and mix, then add the remaining flour mixture. Mix only until no dry streaks of flour remain.
Pour the batter into the prepared pan and spread it into an even layer with a spatula.
Arrange the plum slices on top of the batter in a decorative pattern or scattered evenly.
Bake for 35 to 40 minutes. The cake is done when the top is golden and a wooden skewer inserted into the center comes out clean.
Let the cake cool in the pan on a wire rack for 15 minutes. Run a knife around the edge, then carefully invert the cake onto the rack to cool completely. Dust with confectioners' sugar before serving.