Preheat oven to 325°.
To prepare cake, lightly spoon flour into dry measuring cups; level with a knife. Combine flour and next 4 ingredients (flour through nutmeg) in a large bowl; stir with a whisk. Add apple and pecans; toss gently to combine. Make a well in center of mixture. Combine granulated sugar, applesauce, 1 teaspoon vanilla, and eggs in a bowl; beat with a mixer at medium speed until well-blended. Add to flour mixture. Stir just until moist. Spoon batter into a 13 x 9-inch baking pan coated with cooking spray. Bake at 325° for 55 minutes or until a wooden pick inserted in center comes out clean. Cool completely on a wire rack.
To prepare icing, beat butter and cream cheese with a mixer at medium speed until smooth. Add powdered sugar and 1 teaspoon vanilla; beat just until blended. Spread icing over cake.