Preheat oven to 375°F.
Heat butter in skillet over moderate heat until foam subsides. Reduce heat to low and sprinkle brown sugar evenly over butter, then cook, undisturbed, 3 minutes (not all of sugar will be melted). Remove skillet from heat and arrange apricot halves, cut sides down, close together on top of brown sugar.
Make cake batter:
Sift together flour, baking powder and soda, and salt into a small bowl.
Beat together butter, sugar, and extracts in a large bowl with an electric mixer at medium speed until pale and fluffy, 2 to 3 minutes in a standing mixer or 3 to 4 minutes with a handheld. Beat in eggs 1 at a time, then beat until mixture is creamy and doubled in volume, 2 to 3 minutes.
Reduce speed to low and add flour mixture in 3 batches alternately with buttermilk, beginning and ending with flour mixture, and beat just until combined.
Gently spoon batter over apricots and spread evenly.
If your skillet isn't ovenproof, wrap handle with heavy-duty foil (or a double layer of regular foil) before baking. Bake cake in middle of oven until golden brown and a wooden pick inserted in center comes out clean, 40 to 45 minutes.
Wearing oven mitts, immediately invert a large plate over skillet and, keeping plate and skillet firmly pressed together, invert cake onto plate. Carefully lift skillet off cake and, if necessary, replace any fruit that is stuck to bottom of skillet. Cool to warm or room temperature.
Gourmet, July 2003