Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper.
In a medium bowl, sift together the all-purpose flour, unsweetened cocoa powder, baking soda, baking powder, and salt. Set aside.
In a large mixing bowl, whisk together the granulated sugar, brown sugar, mashed bananas, plain Greek yogurt, vegetable oil, eggs, and vanilla extract until smooth and well combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently with a spatula or wooden spoon until just combined. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes. Then transfer it to a wire rack to cool completely before slicing and serving.