In a standing mixer fitted with a dough hook attachment, combine yeast, brown sugar, salt, and water. Stir to combine.
Add in flour, butter, and mashed potato. Turn mixer on low speed and mix for 5 minutes, or until the dough slightly pulls away from the bowl and is nice and stretchy.
Lightly grease a second bowl. Place dough into bowl, cover with a towel or plastic wrap, and allow to rise for 1 hour and 30 minutes in a warm place, until doubled in size.
Meanwhile, combine brown sugar, flour, and cinnamon in a small bowl. Cut apple into tiny cubes.
Grease an 8 inch bread pan with butter. Set everything aside.
Dust a large work surface with flour. Roll dough out into an 8″x18″ rectangle. Sprinkle cinnamon sugar mixture over dough, then place apple cubes evenly on top.
Working from the shorter side, roll dough into a tight log, then place into prepared bread pan with seam side down. Cover again with a towel or plastic wrap and allow to rise for 1 hour more, until the dough rises above the top of the bread pan.
Preheat oven to 350°F. Bake bread for 50-55 minutes, tenting it with aluminum foil after the first 15 minutes. If you have a temperature thermometer, take the bread out once it reaches an internal temperature of 190°F. If you don’t have this, remove the bread once the top is light golden brown and when you knock on it with your fist, it sounds hollow.
Immediately brush with a little butter to give the top of the bread a nice sheen. Allow to cool for at least 30 minutes before removing from bread pan and cutting into slices!