Heat oven to 350° F. Rub an 8 or 9-inch pan with butter and line with parchment. Spread walnut pieces on a baking sheet, place in the oven, and toast for about 10 minutes. Remove from oven and set aside to cool. Turn off oven.
In a medium bowl, sift together the sorghum flour, arrowroot, baking powder, baking soda, and salt. Place one cup of the toasted walnuts in a food processor and process until finely ground. Add to the flour mixture and stir to combine.
In a large bowl, using a mixer, beat the butter, ¼ cup coconut sugar, and maple syrup together for a few minutes until light and fluffy. Add the bananas and mix on medium speed until the bananas are broken down and smooth. Scrape down the bowl as necessary. Add the eggs one at a time and beat until thoroughly combined. Add the Greek yogurt and vanilla and mix well to combine.
Pour in the walnut-flours mixture and mix on low speed until combined. If you can, let batter rest for 30 minutes. Finely chop the remaining 1 cup of walnut pieces.
Scrape the batter into the prepared pan. Scatter the chopped walnuts evenly across the top of the batter, then sprinkle 1 teaspoon of coconut sugar across the walnuts.
Place pan in the oven and bake for 50-60 minutes, until a toothpick inserted in the center comes out clean. Remove pan from oven, place on a rack, and allow to cool in the pan before turning out onto a wire rack. Cake can be kept, well- covered in the fridge, for 5 days.