Preheat the oven to 350°F. Line cookie sheets with parchment paper.
Drain the syrup from the sweet potatoes and place the sweet potatoes in a blender or food processor and process until smooth.
In a large bowl, beat together the butter and 1/2 cup of white and brown sugar until light and fluffy. Beat in the egg and pureed sweet potatoes until well combined.
In a small bowl, combine the flour, baking soda, baking powder, cinnamon, salt, and nutmeg.
Add the dry ingredients to the batter and mmix until well combined.
Drop spoonfuls of dough onto prepared cookie sheet. Dip the bottom of a drinking glass in the remaining 1 tablespoon of sugar and gently press it to the top of the cookies to slightly flatten.
Bake for 12 minutes. Remove from the oven and cool completely before glazing.
Add the butter to a small saucepan set over medium heat. Melt the butter and continue cooking over medium heat, whisking constantly. Butter will foam as it cooks. Cook until the butter turns a golden brown.
Remove from the heat. Stir in the powdered sugar and whisk until you have a smooth glaze.