Preheat the oven to 350 degrees. Grease a 6- or 12-count doughnut pan with butter. If using a 6-count doughnut pan, you will need to bake two batches.
Start by browning the butter, which should take about 5 minutes total. Place the butter in a small skillet over medium heat, and first let it melt. Keep cooking the butter, stirring it with a spatula, until the foam on top begins to subside and little brown specks begin to appear on the bottom of the pan. Once the milk solid specks are a deep golden brown, pour the brown butter into a large mixing bowl and let cool.
In a large bowl, whisk to combine the flour, baking powder, baking soda, salt, cinnamon, cloves, and ginger. Set aside.
Now that the brown butter has cooled, add the sugar and eggs, and whisk until combined. Add the pumpkin puree and milk, and whisk until smooth.
Add the dry ingredients to the wet and stir to combine. The batter will be thick. Fill the doughnut wells about 3/4 full with the batter, making sure to smooth them out well with a spatula. Bake the doughnuts for 16-18 minutes, until a toothpick inserted into the center comes out clean. Flip the doughnuts out of the pan and let them cool completely.
To make the glaze, whisk to combine the confectioner’s sugar, butter, and milk. Dip the cooled doughnuts into the glaze. If the glaze gets too firm from the butter cooling down, microwave for 15 seconds to melt it again. Let the glaze set on the doughnuts for 10 minutes, then the doughnuts are ready to enjoy!