1. Adjust a rack to the bottom third of the oven and preheat the oven to 325 F.
2. In a large bowl, sift together the cake flour, baking powder, and salt. Add the brown sugar and whisk the mixture by hand to combine.
3. In a small bowl, whisk together the yolks, oil, maple syrup, water, and vanilla. Add the liquid ingredients to the dry mixture by hand to combine.
4. In a small bowl, whisk together the yolks, oil, maple syrup, water, and vanilla. Add the liquid ingredients to the dry mixture and briskly stir with a rubber spatula until just smooth. Do not overmix.
5. In the clean bowl of a stand mixer fitted with the whisk attachment, whip the egg whites on medium speed until foamy. Add the cream of tartar and whip on high speed until soft peaks form. Turn the mixer down to medium speed and gradually add the sugar in a steady stream. Kick the mixer up to high speed and whip until the whites just hold firm (not stiff) glossy peaks.
6. Fold a third of the whites into the batter using as few strokes as possible. Add the remaining whites, folding only until evenly incorporated. Lightly fold in the pecans during the last few strokes. Gently pour the batter into the prepared pan and bake until the top springs back when touched or a toothpick inserted in the middle comes out with a few crumbs attached, 50 to 55 minutes.
7. Remove the cake from the oven and let it cool upside down by inverting the cake pan onto its legs. To remove the cooled cake from the pan, slide a long thin knife or spatula along the sides to loosen and knock the pan sharply on a hard surface until the cake drops out.
8. Frost the top and sides with brown butter icing. To cut the cake, use an angel food cake cutter or a serrated knife and a sawing motion. Store in an airtight container for up to 3 days.

Brown Butter Icing

1. Put the sugar in a medium mixing bowl and set aside.
2. Melt the butter in a small saucepan over medium heat. Using a pan with a light-colored bottom will help you keep track of the color. Let the color of the butter darken to a golden brown. Once the butter is dark brown and you begin to smell a nutty aroma, remove the pan from the heat.
3. You can either pour the butter off carefully to leave behind the milk solids or you can keep and use the butter solids.  Pour the butter into the bowl. Whisk until smooth. Stir in the pecans.
4. As the butter cools, the icing will become firmer. If using the icing as a glaze, use it immediately. If you plan to use the icing as a frosting, allow it to cool to a good spreading consistency.