Preheat the oven to 400 degrees F (200 degrees C).
Melt 1/4 cup butter in a 12-inch cast iron skillet over medium-low heat. Stir in brown sugar, pineapple juice, and dark rum. Cook over medium-low heat until sugar is dissolved and mixture is bubbling, about 5 minutes.
Remove from heat and spread pineapple slices in a layer over the brown sugar mixture, completely covering the mixture. Set skillet aside.
Melt 1/2 cup butter in small skillet over low heat until the butter begins to brown and release a nutty fragrance, about 5 minutes. Watch carefully, butter burns easily. Remove from heat and let cool for 10 minutes.
Whisk together flour, cardamom, salt, baking powder, and white sugar in a large bowl. Whisk in egg and cold milk until just combined. Pour in melted butter and stir to mix thoroughly.
Pour batter over the pineapple slices in the skillet; spread evenly to cover.
Bake in preheated oven until the cake begins to bubble around the edges, the top is browned, and a toothpick inserted into the center comes out clean, about 30 to 35 minutes. Let cool in the skillet for 10 minutes.
Loosen the cake from the skillet by running a knife around the inside edge of the cake. Invert a large plate over the top of the skillet and flip over, releasing the cake to show the pineapple slices on top.