Grease an 8" square pan with nonstick spray or softened butter. Line with a long piece of parchment paper extending up and over two opposite sides of the pan to create a sling. (For details on how to make an 8" round cake, see "tips," below.)
To make the batter: In a large bowl, beat together the peanut butter and brown sugar for 30 seconds. Beat in the eggs, followed by the yogurt, then the mashed banana.
Beat in the baking soda, salt, and cinnamon. Gently fold in the flour and scrape the batter into the prepared pan.
Preheat the oven to 350°F and rest the batter while the oven heats.
Bake the cake for 30-35 minutes until a toothpick inserted into the center comes out clean with only a few moist crumbs clinging to it.
Cool the cake in the pan set on a wire rack for about 20 minutes, then turn it out to cool completely.
Make frosting: In a large bowl, cream the butter and sugar until fluffy, then sift in the powdered sugar. Add a splash of milk if needed. Frost the cooled cake with the frosting.
Notes
Recipe Tips and Substitutions
Oat flour substitute - If you can't have oats or oat flour, a nice substitute is sorghum flour which you can swap in by weight for the oat flour. Or, use your favorite 1:1 gluten free flour substitute by weight.
Rest the batter - For the tenderest and most hydrated crumb, let the batter rest for 20 minutes, or at least as long as the oven heats.
Use ripe bananas - A ripe baking banana is very yellow with streaks and spots of brown. It may be going soft, but should not smell like it's rotting. Extra ripe bananas are perfect in cake and bread because they add natural sweetness and flavor.
Try other spices - Add or substitute ground ginger for the cinnamon. Add a pinch of nutmeg or cardamom for a slightly different flavor note.
Make it refined sugar-free - Substitute coconut sugar or maple syrup for the brown sugar.
Go dairy-free - Use your favorite non-dairy yogurt for the Greek yogurt.