Preheat oven to 350°F (175°C) and lightly grease a donut pan.
In a small saucepan over medium heat, melt 3 tablespoons of the butter until it turns golden brown and smells nutty, then set aside to cool.
In a large bowl, whisk together the flour, baking powder, baking soda, salt, and pumpkin pie spice.
In a separate bowl, whisk together the pumpkin purée, granulated sugar, brown sugar, buttermilk, egg, vanilla, and the cooled browned butter.
Fold the wet ingredients into the dry ingredients until just combined.
Pipe the batter into the donut pan and bake for 10-12 minutes until the donuts spring back when touched.
For the glaze, brown the remaining 8 tablespoons of butter in a saucepan until deep amber, then whisk in the powdered sugar and maple syrup until smooth.
Dip the cooled donuts into the glaze and serve.