Preheat the oven to 325ºF. (162°C)
Measure the unbleached flour, whole-wheat flour, baking powder, and cinnamon into a mixing bowl. Whisk until combined.
Combine the mashed banana, peanut butter, carrot, and water in a glass measuring cup. Stir with a fork till smooth and creamy.
Make a well in the flour, and pour in the banana– peanut butter mixture. Mix on medium speed using a paddle attachment until combined.
Turn the dough out onto a slightly floured surface and gently knead until dough is smooth and still soft. Roll the dough ¼-inch thick and cut the biscuits into desired shapes with cookie cutters.
Place the biscuits on a parchment-lined cookie sheet and bake 12– 15 minutes till dry and firm to the touch. Turn the oven off and leave biscuits in another 20– 30 minutes. Remove them from the oven, cool them on a baking rack, and store them in a cookie tin.