Preheat oven to 350° F and line a 8” cake pan with parchment paper. Set aside.
Place butter in a small saucepan set over medium heat. Cook, swirling occasionally for 6-7 minutes, until butter is foamy and fragrant, and browned bits begin to form. Remove from heat and transfer to a heat-proof bowl to cool while you prepare the rest of the ingredients.
In a small bowl, whisk together flour, almond flour, baking powder, salt, and cardamom. Set aside.
In a large bowl, whisk together the two whole eggs, 1/2 cup of sugar, rum, and vanilla extract. Add browned butter and continue whisking until smooth and homogenized. Add dry ingredients to the bowl and whisk gently until no flour streaks remain.
In the bowl of stand mixer or using a hand mixer, whip egg white on medium-high speed until frothy. Add the remaining 1/4 cup of sugar and continue whipping until thick and glossy. The mixture won’t form peaks, but will be very thick and look marshmallowy.
Use a rubber spatula to gently fold the egg whites into the cake batter until completely combined. Pour the batter into the prepared pan and arrange apple slices on top.
Bake for 30-35 minutes, until the edges begin to pull away from the sides of the pan and the top springs back when touched. Let cool for about 10 minutes before removing from pan and continuing to cool.
While the cake is cooling, make the maple tahini cream. In a small bowl, whisk together tahini and maple syrup until smooth. Using a mixer, whip cold heavy cream until soft peaks form, then stream in tahini mixture. Continue mixing until combined. Spoon soft whipped cream over warm apple cake.