Combine the Banana (2) and Brown Sugar (2 tablespoon) in a pot, then mash/liquify the bananas as much as you can with a whisk. Add Bread Flour (3 tablespoon) and mix well.
Set over medium heat and cook until the mixture has thickened. Chill in the fridge until just warm to the touch.
Add the banana mixture to a stand-mixer bowl (or a large bowl if you're using a hand-held mixer with a dough-hook), along with Bread Flour (2 cup), Egg (1), Instant Dry Yeast (1 1/2 teaspoon) and Salt (1/4 teaspoon).
Knead on medium speed until elastic and smooth, approx 3~5 min. The dough should feel tacky but not too wet. If the dough feels dry and has difficulty coming together, add a bit more mashed/liquified bananas.
Now knead in the Unsalted Butter (3 tablespoon) 1 tablespoon at a time, until fully incorporated. Increase the speed to high, and knead for 10 min until extremely elastic, shiny and smooth.
The dough should be able to stretch out slowly without breaking. Cover the bowl with plastic-wrap and let proof in a warm space for 2 hours, or until doubled.
Transfer the dough onto a floured surface, and roll out into 1/3-inch (1 cm) thickness. Cut out as many donuts as you can, then gather the scraps and cut again. You should have about 12~14 donuts and donut-holes.
Transfer the donuts to a lightly-floured baking-sheet, cover with plastic wrap and proof again for 1 hour, until puffy and slightly risen.
To fry the donuts, add enough Canola Oil (to taste) to a frying-pot until it reaches 1 1/2-inch deep. Set over medium-high heat and bring to a temperature of 350 degrees F (175 degrees C) or until it bubbles up immediately around an inserted wooden chopstick.
Gently lower the donuts into the oil, without crowding the pot, and turn frequently until puffed and golden browned. Drain on a cooling rack and repeat with the rest.
To serve, mix Powdered Confectioners Sugar (1/2 cup) together with Ground Allspice (1 teaspoon) and spoon generously over the warm donuts.