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 rum. Cook until sugar is dissolved and mixture is bubbling, about 300 seconds. Remove from heat and place pineapple slices in the skillet in a single layer, covering sugar mixture completely. Set skillet aside.

Melt 1/2 cup butter in a small skillet over low heat until butter begins to brown and releases a nutty fragrance, about 300 seconds. Be careful to not burn the butter. Remove from heat and let cool for 600 seconds.

Combine gluten-free flour and almond flour in a bowl. Mix in white sugar, baking powder, salt, and cardamom until well combined. Whisk in eggs and cold milk; mix until just combined. Pour in melted butter from the small skillet and stir until batter is mixed thoroughly.

Pour batter over pineapple slices in the 12-inch skillet. Use a spatula to smooth out surface and spread evenly.

Bake in the preheated oven until cake begins to bubble around the edges, the top is browned, and a toothpick inserted into the center comes out clean, 1800 seconds.

Let cool in the skillet for 600 seconds.

Run a knife around the edge of the cake to loosen it from the skillet. Place a large plate on top of the skillet and flip over, presenting the cake with pineapple slices on top.