Preheat oven to 350 degrees F. Butter and sugar a 9 by 2-inch round baking pan or 6 ramekins (4-inch diameter).
Melt 1/2 stick butter in a large saute pan over medium-high heat until hot. Heat until butter begins to turn brown. Remove pan from heat, add vinegar and brown sugar, and mix well. (A sauce whisk is a good tool to use here.) Return to heat to melt sugar, if necessary.
Pour a little of the brown sugar sauce in the bottom of the ramekins or baking pan. Cut each pineapple ring into pieces, maintaining the shape of the ring (see Chef's Note). Add 1 pineapple ring to each ramekin or line the bottom of the large pan with pineapple. Add mango around the pineapple.
Drizzle pineapple with remaining brown sugar sauce and sprinkle with chopped nuts.
Make the batter:
In a large bowl, sift together flour, baking powder, and salt. Set aside.
Put 1 stick of softened butter in the bowl of an electric mixer. Beat on medium to high speed with the paddle attachment until light and fluffy. Add 1/2 cup superfine sugar and beat very well, until mixture is white and very light and fluffy. Beat in egg yolks, 1 at a time, beating well after each addition.
Alternately mix 1/3 cup milk, then 1/3 of flour mixture into butter-egg mixture in 3 portions. Do not overmix.
In another bowl, beat egg whites with clean beaters or a balloon whisk. When whites begin to foam, add remaining 1 tablespoon sugar and beat until they hold soft peaks. Scrape 1/2 the whites into batter and mix gently until blended. Fold in remaining whites carefully, until just blended.
Divide batter among ramekins or pour into baking pan. (If using ramekins, place them on a baking sheet.) Place cake in preheated oven. Bake until a tester inserted into the middle comes out clean, about 20 to 30 minutes for individual cakes or about 50 minutes for the large cake. Remove from oven and let cool about 10 minutes in the pan. Run a knife around the edges to loosen the cake(s) and invert onto deep dessert plates or a deep platter (otherwise, the sweet syrup may overflow the plate!).
Upside-down cakes are old-fashioned American desserts. In summer, use ripe peaches and nectarines, even plums. I added mango for a twist on the classic pineapple upside-down cake. The cake is uncomplicated and not very sweet by itself; the pineapple juices and brown sugar sauce permeate the cake as it bakes. The result is luscious. It does not matter whether you use light or dark brown sugar. It depends on your preference and what you might have in your cabinet. Some people do not care for the stronger molasses flavor of dark brown sugar; others are disappointed in the flavor of light. It makes no difference to the cake. I suggest cutting the pineapple ring into pieces because whole rounds can be hard to cut when baked. The ring shape is a decorative touch to your pineapple upside-down cake, but it is not essential.