Step 1Melt the butter in a medium saucepan over medium heat. Cook until the solids separate and begin to brown to a dark golden color, 7 to 10 minutes. Remove it from the heat and let it cool. Don't chill. It needs to remain in liquid form. Set aside.
Step 2Heat the oven to 350 degrees. Place the rack in the center of the oven. Spray a 10- by 2-inch round cake pan with nonstick spray. Line with parchment paper, then coat the paper lightly with spray.
Step 3Sift together the almond flour, flour, powdered sugar, baking powder, baking soda and salt into the bowl of a mixer fitted with a paddle attachment, or use a large bowl and a hand mixer. Add the cinnamon, ginger, nutmeg and brown sugar. Mix the dry ingredients at low speed for 30 seconds.
Step 4Add the egg whites and mix on medium speed for 3 minutes. Add the pumpkin puree and orange zest and mix for 30 seconds.
Step 5Add the melted butter. Be sure to scrape in all the browned bits from the bottom of the pan. Mix for 30 seconds on medium speed, then turn the mixer to high speed and mix for 3 minutes more, scraping down the sides of the bowl. Financier batter will keep in the refrigerator for up to 2 weeks. (When using refrigerated batter, be sure to bring it to room temperature, then stir the mixture from the bottom up to reincorporate any butter that might have separated and sunk to the bottom. Beat the batter by hand or with a mixer to warm it up and mix it well.)
Step 6Pour the batter into the prepared pan. Bake for 15 minutes. Rotate the cake for even browning and bake for 10 minutes more, or until a toothpick inserted in the center comes out clean. Let cool in the pan on a rack for 5 minutes, then invert it onto the rack, remove the cake pan and parchment, and cool completely. Wrapped airtight, the financier will keep at room temperature for up to 2 days or in the freezer for up to 2 weeks.