In a large bowl, combine the brown sugar, vanilla extract, butter, pumpkin puree and egg. Beat with an electric mixer until well blended.
Next, add the dry ingredients, flour, baking soda, ginger and cinnamon, to form soft dough. Stir to combine until no white flour streaks are showing.
Cover the bowl with plastic wrap and refrigerate for about 20 minutes so that your cookies will have a nice deep flavor.
Preheat the oven to 375°F. Then, line a baking sheet with parchment paper or a silicon baking mat.
Remove the dough from the fridge and roll into 1-inch balls and put onto the prepared baking sheet. Slightly flatten each ball with the round side of a spoon or the bottom of a glass.
Bake for 7-10 minutes or until lightly golden brown. These cookies do not spread much at all, so I ended up baking about 20 on a tray. Once baking is done, remove from tray immediately and place on cooling rack.
Browned Butter Frosting: In a small saucepan, heat the 1/3 cup butter over medium heat, stirring constantly. Cook until butter turns a nice golden brown. It took mine about 5 minutes and when it turned, it turned very suddenly. Once that happens, remove from the heat.
Next, add in the powder sugar, vanilla extract and milk. Stir well until frosting is smooth and spreadable.
Let cookies cool for about 20 minutes before frosting, you do not want the frosting to melt off the cookie. Serve immediately and enjoy!