Place butter in a saucepan over medium heat, stirring until it turns golden brown and smells nutty, then set aside to cool slightly.
Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, oats, baking soda, pumpkin pie spice, cinnamon, and salt.
In a large bowl, beat the browned butter with the brown sugar until well combined.
Add the egg, pumpkin puree, and vanilla extract to the butter mixture, whisking until smooth.
Gradually stir the dry ingredients into the wet until just combined.
Scoop rounded tablespoons of dough onto the prepared baking sheet.
Bake for 12 to 14 minutes, or until the edges are set.
Allow cookies to cool completely on a wire rack.
Whisk together powdered sugar, milk, and vanilla extract until smooth, then drizzle over cooled cookies.