In a medium saucepan combine the apples, mulberries, maple syrup, and 2 tablespoons of the water. Bring to a boil, reduce heat, cover and cook for about 5 minutes, stirring occasionally, until the apples are softened and beginning to break down.
Remove the lid and stir in the arrowroot flour, cinnamon, and 1 tablespoon of water. Cook and stir for 1 minute more, until slightly thickened. Remove from heat and let cool slightly.
To assemble the buns, spread the almond butter evenly over the rolled out dough. Spoon the cooled apple mixture on top and spread evenly. Roll up dough, let rise and bake as directed.
(Optional: About 5 minutes before the buns are finished baking, remove them from the oven and brush with an additional 1 tablespoon of maple syrup. Return to the oven to set the glaze.)