Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking pan or line it with parchment paper for easy removal after baking.
In a large bowl, cream together the softened butter and brown sugar until light and fluffy. Add in the egg and vanilla extract followed by the pumpkin puree, mixing until well combined.
In another bowl, whisk together all-purpose flour, ground cinnamon, ground ginger, ground nutmeg, baking soda, and fine sea salt until evenly mixed.
Gradually fold the dry ingredients into the wet mixture until just combined; be careful not to overmix as this can lead to dense bars.
Pour the batter into your prepared pan and spread it evenly using a spatula. Bake in preheated oven for about 25 minutes or until a toothpick inserted into the center comes out clean.
While your bars are cooling down completely in their pan, prepare the frosting by beating softened butter and brown sugar together until creamy; then add powdered sugar, cinnamon, salt, vanilla extract, and milk until you reach desired consistency.