Preheat the oven to 350 degrees F.
Melt 4 tablespoons of the butter in a 10-inch heavy-bottomed, ovenproof skillet set over medium heat. Add the brown sugar and stir to combine. Cook for 6 minutes. Then add the pineapple and cook for 1 more minute. Remove the skillet from the heat.
In a medium bowl, whisk together the flour, baking powder, ginger, cinnamon, and salt.
In the bowl of an electric mixer fitted with the paddle attachment, beat together the remaining 8 tablespoons butter and granulated sugar on medium speed until the mixture is light yellow and fluffy, about 3 minutes. With the mixer running on low speed, gradually add the eggs, vanilla, and rum. Mix for 1 minute.
With the mixer running on low speed, mix in a third of the flour mixture and half of the sour cream. Add another third of the flour mixture and the remaining sour cream. Remove the bowl from the mixer, and fold in the remaining flour mixture. Pour the batter over the pineapple in the skillet, and spread it out evenly.
Bake, rotating the skillet halfway through, for 50 minutes, or until a cake tester inserted in the center of the cake comes out clean. Let the cake cool in the skillet for 20 minutes. Run a sharp knife around the edges of the cake to loosen it from the skillet. Carefully invert the skillet over a serving plate, releasing the cake. Sprinkle the salt over the top of the cake.
Hawaiian sea salt is available in many specialty food stores. However, if you cannot find it, kosher salt is an acceptable alternative.
Reprinted with permission from One Girl Cookies by Dawn Casale and David Crofton, Clarkson Potter copyright (c) 2012