In a medium bowl, whisk together the flour, baking soda, salt, and nutmeg. Set aside.
In a big bowl, beat together the white sugar and butter until pale and fluffy. Scrape down sides of bowl, then mix in vanilla and egg, followed by the pumpkin puree. Slowly stir in the dry ingredients.
Cover dough and chill in fridge for at least two hours; overnight is fine.
Preheat oven at 350-degrees. Line two baking sheets with parchment paper. Use a tablespoon scoop to place dollops of batter spaced out on the sheets; the dough will be sticky to work with.
Bake cookies until golden brown, about 13 minutes. Let cool completely on sheets.
To make the maple glaze, whisk together the confectioners' sugar, butter, maple flavor, and a tablespoon of water. Add more water or sugar, as needed, to reach a thick yet spreadable consistency. Use the back of a spoon to swirl a thin layer of glaze atop each cookie.
Store cookies, between layers of parchment or wax paper, in a sealed container up to 3 days.
OM NOM NOM!