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 flour, sugar, baking soda, salt, cardamom, ginger, and cinnamon.
In a medium bowl, whisk together eggs, vanilla extract, and yogurt. Whisk in the mashed bananas and maple syrup. 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 batter into prepared pan. Bake for 35-45 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.
When cake is cooled, make frosting: In a large bowl using an electric beater, whip the butter. Add the chocolate and sugar a ½ cup at a time, continuing to beat until a creamy frosting forms. Taste for sweetness and add more sugar if you wish. Frost the top of the cake with the frosting.