Preheat oven to 350°.
Coat bottom of a 9-inch round cake pan or 9-inch cast iron skillet with melted butter. Sprinkle with brown sugar. Top with plums, arranging in a circular pattern; set aside.
Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, baking powder, baking soda, and salt in a large bowl. Beat softened butter and sugar with a mixer at medium speed until blended. Add orange rind, vanilla, and egg, beating until well blended. Add flour mixture to sugar mixture alternately with buttermilk, beating well after each addition. Pour batter over plums.
Bake at 350° for 35 minutes or until a wooden pick inserted in center comes out clean. Cool in pan 5 minutes on a wire rack. Loosen cake from sides of pan using a narrow metal spatula. Place a plate upside down on top of cake pan; invert cake onto plate. Serve warm.