Dissolve 1 teaspoon of the sugar in 75 ml of lukewarm water, then sprinkle in the dried yeast and set aside for 15 minutes, or until it is frothy.
Sift the two flours, allspice, cinnamon and remaining sugar into a large bowl, adding any bran left in the sieve. Mash the bananas into a smooth paste, then stir them into the spiced flour with the egg and egg white. Pour the yeast mixture into the centre. Stir to form a dough, adding a little extra flour if the mixture is too wet.
Turn out the dough onto a lightly floured surface and knead for 5 minutes, or until smooth.
Grease a bowl with a little oil and roll the dough around so that it is lightly coated with oil. Loosely cover the bowl with a moist cloth and leave the dough to rise in a warm place for 2 hours, or until it has doubled in size. Meanwhile, lightly grease a loaf tin measuring about 19 × 12 × 9 cm (7½ × 4½ × 3½ in) and set aside.
Turn out the dough onto a lightly floured surface. Punch it down, knead for 1-2 minutes, then roll it into shape and place it in the tin. Cover it loosely and leave to rise for 45 minutes, or until it reaches the top of the tin. Heat the oven to 200°C.
Bake the bread for 40 minutes, or until brown on top and a skewer comes out clean. Stand the tin on a wire rack for 20 minutes before turning the loaf out. Serve it spread with your choice of topping.