Place a rack in the center of the oven and preheat to 350°F. Grease a 9-inch cake pan with butter and line with a round of parchment paper.
Melt butter in a saucepan over medium heat. Butter will begin to foam and crackle as it melts. When the crackling subsides, the butter will begin to brown. Swirl the pan as the butter cooks. When the butter browns and begins to smell nutty, remove the pan from the flame and transfer the butter to a small bowl. Taking the butter out of the hot saucepan will stop the butter from overcooking and burning. Set aside to cool.
In a large bowl, whisk together the oat flour, sugar, baking powder, salt, cardamom, and ginger.
In a medium bowl, whisk together the eggs, vanilla extract, yogurt, and bananas. When butter has cooled, whisk in the browned butter.
Add the wet ingredients all at once to the dry ingredients. Fold together, making sure to scrape the bottom of the bowl.
Pour the batter into the prepared pan and rest it for 15 minutes. Then bake for 45 minutes or until a toothpick inserted into the center of the cake comes out clean. Remove from the oven and allow to rest in the pan for 15 minutes before inverting onto a wire rack to cool completely.
Make Frosting
In a heatproof bowl, combine the butter and chocolate. Set the bowl over a pot of simmering water, and cook until the butter and chocolate is melted, stirring if necessary. Stir very well to make sure they are combined and smooth, then let cool. When both the cake and icing are cool, cover the top of the cake with the icing.
Notes
Rest the batter - Let the batter rest for about 15 minutes before baking it. This will help create that lovely, fluffy crumb.
Bake in an 8-inch pan - For a thicker, even fluffier cake, feel free to bake the batter in an 8-inch cake round.
Can my bananas be too ripe? Not really! The riper bananas are, the better they become for baking, bringing loads of natural sweetness and flavor to your bakes. That being said, if you notice any mold on your bananas, toss them straight into the compost and do NOT use them!
Add nuts - Scatter sliced, toasted (or not) almonds across the top of the icing before it fully sets.
Skip the icing - For a super simple cake, simply sift a little powdered sugar over the top of the cake instead of the chocolate icing.
Other spices - Play around with the spices here—swap cinnamon for the cardamom, add a pinch of nutmeg, or try adding 1 tablespoon of peeled, grated fresh ginger.