Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper if desired.

In a large mixing bowl, combine the softened butter and granulated sugar. Beat until light and fluffy, about 2-3 minutes.

Add in the canned pumpkin, egg, and vanilla extract. Mix until fully incorporated.

In a separate bowl, whisk together the flour, baking soda, baking powder, cinnamon, nutmeg, ginger, and salt.

Gradually add the dry ingredients to the wet mixture, mixing until just combined. Do not overmix.

Drop spoonfuls of dough onto the prepared baking sheets, spacing them about 2 inches apart.

Bake in the preheated oven for 10-12 minutes or until the edges are set and the tops are just firm to the touch. Allow cookies to cool on the baking sheet for 5 minutes before transferring them to a cooling rack.

While the cookies are cooling, make the brown butter icing. In a small saucepan over medium heat, melt the butter. Continue cooking the butter until it becomes golden brown and has a nutty aroma, taking care not to burn it. Remove from heat and let cool slightly.

In a separate bowl, whisk together the powdered sugar, milk, vanilla extract, and a pinch of salt. Slowly drizzle in the brown butter while whisking until smooth and the desired consistency is achieved.

Once the cookies are fully cooled, drizzle or spread the brown butter icing over each cookie. Allow the icing to set before serving.