Preheat oven to 350ºF. Grease two 9-inch round cake pans or a 13×9-inch baking pan.
Combine flour, pecans, baking powder, cinnamon, nutmeg and salt in a medium mixing bowl. Stir well and set aside. Whisk together eggs, oil, sugar, applesauce and vanilla. Stir in carrots until well blended. Add carrot mixture to flour mixture and stir with a spatula until combined.
Pour batter into prepared baking pan(s). Bake 20 to 25 minutes if using the round pans or 30 to 35 minutes in the larger pan, or until wooden pick inserted in center comes out clean. Cool round cakes in pan for 10 minutes then turn out on wire rack to cool completely.
For the Frosting:
Set up a double boiler; bring 1 to 2 inches of water to a boil in a large saucepan or double boiler. If improvising double boiler, make sure a medium heat-proof bowl (8-cup capacity) will fit on the top of the saucepan without the sides hanging too far over the edges. Reduce the heat to maintain a simmer.
Beat egg whites, maple syrup and vinegar in the medium heat proof bowl with an electric mixer.Place the bowl over the simmering water, and beat until the mixture is glossy, fluffy and the eggwhites hold medium peaks, about 7 minutes. Note: it may be necessary to adjust the heat under the bowl to keep a steady simmer.
Remove the bowl from the heat, and continue beating until the mixture is cooled, 3 to 5 minutes. Beat in the vanilla and salt.
For the layer cake, place one cake on cake plate; spread frosting on top surface (only). Top with second layer and repeat. Sprinkle cake top with additional chopped pecan, if desired. Serve immediately or cover and refrigerate.