Make the brown butter by placing the butter in a heavy-bottom saucepan over medium heat.
Once the butter melts, continue to cook it over medium heat, occasionally swirling it. It will foam then it will crackle and pop. At this point, the water is being cooked out. Once the popping and crackling stops, this is when the milk solids will begin to brown and cook. Keep a close eye on it so that it doesn't burn. It is cooked once it's a light brown.
Immediately pour the browned butter into a bowl, and scrape out the browned milk solids that are in the pan as well.
Place the browned butter aside to cool while assembling the rest of the cake.
Preheat the oven to 350 degrees F
Line the bottoms with parchment paper and grease 3, 6-inch round pans.
In a medium bowl, sift the flour, baking soda, baking powder, salt, cinnamon, ginger, nutmeg, and allspice. Set aside.
In the bowl of a mixer fitted with a whisk attachment, beat the egg whites until soft peaks form.
Place the egg whites in a bowl and set aside.
In the bowl of a mixer fitted with a paddle attachment, combine the cooled brown butter, sugar, and eggs. Beat on medium-low until just combined (1 or 2 minutes).
Slowly add in the dry ingredients. Mix until just combined.
Mix in the pumpkin puree and fold in the beaten egg whites.
Divide the cake batter evenly between the three pans.
Bake for 35 to 40 minutes or until a toothpick inserted in the center comes out with moist crumbs attached.
Place the cake pans on a cooling rack and allow the cakes to cool in the pans for about 15 minutes.
Turn the cakes out on to the cooling racks and allow them to cool completely.