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Preheat oven to 350 degrees.
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Spray an 8-inch square baking pan with vegetable cooking spray, and set aside.
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In a medium bowl, combine flour, cocoa, and salt. Mix well.
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In a large bowl, whisk together egg whites, egg, sugar, applesauce, oil, and vanilla.
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Stir in flour mixture until just blended; do not overmix, or brownies will be dry.
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Pour batter into prepared pan, and sprinkle with walnuts.
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Bake until just set, and a toothpick inserted in center comes out clean, about 25 minutes.
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Place pan on a wire rack and cool for at least 15 minutes.
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Cut brownies into squares.
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Variation: For slightly chewier brownies, replace the 1/2 cup of cake flour with 7 Tbl. of all-purpose flour. (Sifting is not necessary.) Proceed with the recipe as directed.
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For extra easy cleanup: line the baking pan with waxed paper, letting paper overhang the edges of the pan by 1 inch, before pouring in the batter. Spraying the pan with oil is not necessary.
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Bake and cool as directed, then simply lift them out of the pan by grasping the edges of the waxed paper.