Preheat oven to 350°F. Spray donut pan with non-stick spray and set aside.
Whisk the flour, baking soda, pumpkin pie spice, and salt together and set aside.
In a large bowl, whisk the brown butter, sugar, brown sugar, eggs, pumpkin, and vanilla together. Pour in half of the dry ingredients and stir until combined. Then add in the milk and the remaining dry ingredients Mix until fully combined and smooth.
Transfer the batter to a piping bag, then cut of the tip and pipe into the donut pan, filling about halfway. You can also just spoon the batter into the pan if desired.
Bake at 350°F for 10-11 minutes or until the edges and tops are lightly browned and a cake tester comes out clean. Invert the donut pan on a cooling rack, and the donuts should slide right out!
In a small bowl, make the glaze by whisking together the powdered sugar, vanilla, and melted brown butter. Add in the milk 1 teaspoon at a time until your desired consistency is reached.
Dip the donuts into the glaze and place back on the cooling rack so the excess can drip off. The donuts are easy to glaze if they are cool, but they taste amazing when they're warm, so it's up to you!
Notes
Make the brown butter: In a small saucepan, melt the butter over medium heat. Once melted, stir frequently and watch closely for the formation of brown specs on the bottom of the pan and a nutty aroma. Once you see brown specs, stir for just a few more seconds (it browns super quickly). Remove from the heat and immediately pour into a different bowl to prevent it from burning. Let the brown butter cool slightly (5-10 minutes) before using. Note: For every ½ cup (1 stick or 113g), you'll lose about 1 tablespoon (14g) to evaporation while cooking. So you can either add in an extra 1 tablespoon before browning, or if you're browning big batches, just make up the weight difference in your recipe with regular unsalted butter if needed.Store in an airtight container at room temperature for up to 3 days.