Preheat oven to 400 degrees F (200 degrees C). Place bananas on a baking sheet and prick peels with a fork.
Bake bananas in the preheated oven until peels turn black and flesh is very tender and mushy, 15 to 20 minutes. Remove from oven and let cool to room temperature, about 20 minutes.
Reduce oven temperature to 350 degrees F (175 degrees C). Lightly grease a loaf pan.
Split banana peels with a sharp knife and transfer flesh to a large bowl. Add honey and oil; beat with an electric mixer on low speed until combined. Beat in egg yolks 1 at a time. Add walnuts, cinnamon, nutmeg, and allspice to the batter.
Beat egg whites with salt in a glass, metal, or ceramic bowl until soft peaks form.
Fold beaten egg whites into the batter. Pour into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes. Invert onto a plate and let cool.