Preheat the oven to 350°F. Butter a 9x9 square metal baking dish and line with parchment paper.
In a large bowl, cream the butter and granulated sugar until light and fluffy. Add the eggs and beat on medium speed for 2-3 minutes until light and bubbly.
Add the mashed bananas, yogurt, and vanilla extract and beat to combine. Add the baking soda and salt and combine.
Gently fold in the flour with a large spoon until just combined. Do not overmix. Pour the batter into the prepared baking dish.
Bake at 350°F for 30 minutes. The banana cake is done when it is a medium golden brown and a toothpick inserted into the center comes out clean.
Meanwhile, make the frosting: In a medium bowl add the chocolate chips. Microwave for 1 minute at 50% power, stirring and repeating as needed until melted.
In a large bowl combine the softened cream cheese, softened butter, powdered sugar, melted bittersweet chocolate chips, and vanilla extract.
Once the banana cake has fully cooled, spread the chocolate frosting on top. To cut clean slices run a knife under hot water, dry the knife and cut a slice, repeating as needed.
Notes
Bittersweet Chocolate Chips: Use high-quality chocolate chips, which melt better than store brands or lower-quality brands. I recommend Ghirardelli or Guittard which can be found at many grocery stores, Target, or Whole Foods. You can also use dark chocolate chips or bars.
Use Full-Fat Yogurt or Sour Cream: Yogurt or sour cream can be used interchangeably. They both add creaminess and fat which adds moisture to the cake. That fat also adds flavor and richness.
Whip the Eggs: Unlike traditional banana bread, in this Banana Cake eggs are whipped to add air bubbles, then the flour is gently folded into the whipped egg batter. This gives the cake a lighter texture than banana bread, which tends to be denser. It also helps to have the eggs at room temperature, which is less important in banana bread.
Extra Vanilla: I add extra vanilla extract (1 ½ teaspoons) to give the cake a little extra boost of vanilla flavor that is common in so many cakes, like birthday cakes.