Preheat the oven to 375 degrees F (190 degrees C).
Butter an 8-inch cast iron or oven-proof skillet. Dust with flour, tapping out any excess.
Whisk together 1 cup flour, salt, baking powder, and baking soda in a medium bowl.
Combine 4 tablespoons butter and 3/4 cup sugar in a large bowl; beat with an electric mixer on medium speed until pale and fluffy. Beat in egg. Add flour mixture in 3 batches, alternating with buttermilk, beating batter briefly after each addition. Pour batter into the prepared skillet and smooth the top with an offset spatula. Fan plum slices on top of batter, and sprinkle with remaining 2 tablespoons sugar.
Bake in the preheated oven until golden brown and a toothpick inserted into the center comes out clean, 35 to 40 minutes. Let cool slightly before serving.