1/2 medium pineapple, peeled, quartered lengthwise, and cored
3/4 stick unsalted butter
3/4 cup packed light brown sugar
1 1/2 cups all-purpose flour
2 to 3 teaspoons ground cardamom
2 teaspoons baking powder
1/4 teaspoon salt
3/4 stick unsalted butter, softened
1 cup granulated sugar
2 large eggs
1 teaspoon vanilla
1 tablespoon dark rum
1/2 cup milk
Preheat oven to 350°F.
Cut pineapple crosswise into 3/8-inch-thick pieces. Melt butter in skillet. Add brown sugar and simmer over moderate heat, stirring, 4 minutes. Remove from heat. Arrange pineapple on top of sugar mixture in concentric circles, overlapping pieces slightly.
Sift together flour, cardamom, baking powder, and salt. Beat butter in a large bowl with an electric mixer until light and fluffy, then gradually beat in granulated sugar. Add eggs, 1 at a time, beating well after each addition. Beat in vanilla and rum. Add half of flour mixture and beat on low speed just until blended. Beat in the milk, then add remaining flour mixture, beating just until blended. (Batter may appear slightly curdled.)
Spoon batter over pineapple topping and spread evenly. Bake cake in middle of oven until golden and a tester comes out clean, about 45 minutes. Let cake stand in skillet 5 minutes. Invert a plate over skillet and invert cake onto plate (keeping plate and skillet firmly pressed together). Replace any pineapple stuck to bottom of skillet. Cool on plate on a rack.
Serve cake just warm or at room temperature.
I worked with this recipe.