Heat oven to 350°F. Grease a 9-inch pan with butter and line with a circle of parchment and set aside. Spread 2 cups walnut pieces on a baking sheet, place in the oven, and toast for about 10 minutes. Remove from oven and set aside to cool.
In a medium bowl, whisk together the 1 ¾ cups oat flour, 1 tablespoon baking powder, and ½ teaspoon fine sea salt Place one cup of the toasted walnuts in a food processor and process until finely ground. (Set aside the remaining walnuts.) Add to the flour mixture and stir to combine.
In a large bowl, using a mixer, beat the ½ cup unsalted butter, ½ cup brown sugar and ½ cup granulated sugar together for a few minutes until light and fluffy. Add the 3 bananas and mix on medium speed until the bananas are broken down and smooth. Scrape down the bowl as necessary. Add the 2 large eggs one at a time and beat until thoroughly combined. Add the ⅓ cup Greek yogurt and 1 teaspoon pure vanilla extract and mix well to combine.
Beat in the flour mixture on low speed until combined. Pour the batter into the pan and rest for 15 minutes. Finely chop the remaining 1 cup of walnut pieces and scatter them evenly over the top of the cake.
Place the pan in the oven and bake for 50-60 minutes, until a toothpick inserted in the center comes out clean. Place the cake on a wire rack to cool, and cool in the pan for 20 minutes before turning out to cool completely.
Notes
Other flour. If you don't need to eat gluten-free, feel free to swap the oat flour for all-purpose flour, whole wheat flour, or a mix of the two. I also like to use my homemade gluten free flour blend or 1:1 flour blend for a slight change. You'll need 1 ½ cups of wheat flour or 1 ¾ cups of gluten-free flour if you go this route.
Use ripe bananas. For this cake, use the ripest bananas you've got. You know, the ones no one wants to eat and which have been kicking around the kitchen counter for longer than you'd like to admit. Those are perfect.
Can I use frozen bananas? Absolutely! Just make sure to thaw and drain off any water before adding the bananas to the batter.
Don't worry if the batter "breaks."The batter gets a little funny-looking when you beat in the bananas - not to worry! This is normal. As you continue to add ingredients it will turn into a smoothly cohesive batter.
Make it dairy-free. Substitute coconut oil or your favorite non-dairy butter for the butter, and non-dairy yogurt for the yogurt.
Another nut. Instead of walnuts, try another nut of your choice, such as almonds or pistachios
To serve. At breakfast, with a hot cup of coffee! Or as dessert, with whipped cream or vanilla ice cream.