Stir together the butter, brown sugar and granulated sugar in a large bowl until the sugar has dissolved.
Add the egg, vanilla and pumpkin puree, stirring until just combined.
Add the flour, pumpkin spice, baking soda and salt and mix until the ingredients are incorporated.
Fold in the oats.
Scoop three tablespoons worth of dough into balls and place on a parchment paper lined cookie sheet.
Bake at 350F for 11-12 minutes or until no longer shiny on top.
Remove from the oven and allow to cool on the cookie sheet.
In a small saucepan, cook the butter over low-medium until browned.
Pour into a small bowl and allow to cool about 10 minutes.
Add the powdered sugar, whisking until all lumps have dissolved.
Drizzle over the cooled cookies.
Notes
Room temperature ingredients: Starting with room temperature ingredients is what ensures the batter is the right consistency to produce crispy cookies.
Bringing eggs to room temperature: If you forgot to take the egg out in advance, you can quickly bring it to room temperature by placing it in a bowl of hot water for 5 minutes.
Bake immediately: While many kinds of cookie dough can be refrigerated prior to baking, this dough is just the right consistency at room temperature, which is why I recommend baking them immediately.
The importance of leaving cookies on the baking sheet: Removing the cookies when they're no longer shiny means they'll be slightly under-cooked. Leaving them on the hot baking sheet allows them to finish baking without continuing to brown.
Doneness of browned butter: You know your butter is adequately browned when it develops brown flecks and a nutty smell.
Salvaging overcooked browned butter: Should you overcook your butter, you can pour it through a strainer to remove the burned flecks.