Preheat an oven or toaster oven to 350ºF.
Drain and discard syrup from yams.
Add yams, melted butter, maple syrup, brown sugar, and cinnamon to a large bowl.
Mash all ingredients together and beat into a smooth paste with an electric beater
Snip a small tip off the corner of a food storage or freezer sized ziplock bag and fill with yam mixture.
Remove Fillo shells from package and place on a foil lined baking sheet (for easy cleanup).
Using a plastic bag filled with yam mixture, pipe out enough mix to fill each shell.
Place baking sheet with prepared shells into preheated oven for 10 minutes to warm filling.
Remove shells from oven and carefully top each "pie" with four to five mini marshmallows.
Turn oven to broil, move rack to highest setting, and return pies topped with marshmallows to the oven.
Browning marshmallows will vary depending on your oven but it will happen VERY FAST.
Once your marshmallows are an attractive, golden-brown color, you're done.
Remove the pies from your oven and allow to cool a few minutes before plating them.