- Preheat oven to 400 degrees
- In a heavy-bottomed saucepot, melt butter and allow to cook until butter separates. Continue cooking until milk solids fall to bottom of the pot; toast to a golden brown color with the butter smelling a bit like hazelnuts. Remove butter from heat and carefully stir to disperse toasted butter solids.
- Very carefully strain butter through a fine sieve into a temperature-resistant container and discard solids.
- Sift together powdered sugar, almond flour, all-purpose flour, salt, cinnamon, and cardamom.
- Place into bowl of a standing mixer fitted with whisk attachment. Turn mixer on the lowest speed and carefully stream in egg whites and orange zest. Allow dry ingredients to be fully incorporated and increase mixer speed. Mix batter until uniform and smooth.
- Reduce mixer speed back to low and carefully stream in the room-temperature brown butter. Incorporate until smooth.
- In a heavy-bottomed saucepot cook sugar, water, and lemon juice until dark amber in color. Remove pot from heat and carefully whisk in rum, followed by the butter.
- Heat a well-seasoned 9-inch cast iron pan over high heat. Place sliced figs cut side down in pan followed by one tablespoon of butter.
- As butter begins to froth and brown, pour in ¼ cup of rum caramel; allow it to spread out evenly over the pan. Turn off heat and add cake batter to pan.
- Place the pan immediately into the preheated 400-degree oven. Let it bake for 15−20 minutes or until a cake tester comes out clean.
- Remove cake from the oven and let rest for 5 minutes. Use a thin knife to loosen the sides of the cake from the pan. Place a dinner plate inverted over the cast iron pan and very carefully flip it. Serve warm with vanilla ice cream and extra spiced rum caramel.