Wash yams thoroughly and poke on all sides with a fork to allow steam to escape while cooking.
Arrange yams in a spoke pattern on several layers of paper towel in the microwave.
Cook on the "baked potato" setting or until yams are soft all the way through when poked with a fork.
Remove from microwave oven and allow to sit and cool for 10 minutes or until they are cool enough to handle.
Slice yams in half lengthwise and scoop out cooked contents with a spoon. Discard skins.
In a bowl, combine yams, butter, maple syrup, vanilla extract, ¼ cup of brown sugar, and heavy cream. Mix well, mashing with a fork or potato masher until most lumps are smooth.
Spread yam mixture into a 9x13 glass baking dish or similar casserole dish.
Sprinkle mini marshmallows evenly over the yams.
Bake at 375 degrees for 20-25 minutes or until the marshmallows are toasted brown. Remove from oven and serve immediately.