For the Cake: In the bowl of a stand mixer, combine eggs, sugar, and salt. Using a whisk attachment, whip on high until eggs become thick, pale, and ribbon-y, about 10 minutes. With mixer running on medium, slowly drizzle in oil in a thin stream until fully incorporated and thick. Whisk in grated ginger, about 10 seconds, then shut off mixer and remove bowl.
Using a flexible rubber spatula, fold in almond flour, walnuts, cinnamon, and nutmeg until combined. Fold in grated apple. Pour batter into prepared pan and gently tap on the counter to evenly distribute. Bake until golden brown and top of cake bounces back to the touch, about 50 minutes.
While cake is still warm, loosen exposed short sides of cake from pan using an offset spatula or butter knife. Using parchment paper as a handle, lift cake out of pan and onto a cutting board. Cut cake into diamond shapes by making a series of diagonal slices in one direction across the cake, then making another series of diagonal slices in the opposite direction. Lift sliced cake, again using parchment, and return to pan. Strain syrup over warm cake using a fine-mesh strainer, then let sit at least 1 hour to allow cake to fully absorb syrup. Serve.
Special equipment
9-by-13-inch anodized aluminum baking pan, medium nonreactive saucepan or saucier, stand mixer, instant-read thermometer, fine-mesh strainer
