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In a medium saucepan over medium-high heat, cook applesauce until just beginning to boil. Stir in oatmeal. Set aside; let stand for 20 minutes.
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Preheat oven to 350 degrees F. Make sure oven rack is at least 6" from top of oven. Coat a 9" x 9" baking dish with non-stick cooking spray; set aside.
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In a medium bowl, combine flour, baking soda, cinnamon, nutmeg, and salt. Set aside.
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In a large bowl, cream together butter and sugar until light and fluffy, about 2 minutes. Add vanilla and eggs, mix until combined. Add flour mixture to butter mixture in two additions; mix until just combined after each addition. Add applesauce mixture and mix until just combined.
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Pour batter into prepared baking dish. Bake for 40 to 50 minutes, until cake has browned and a toothpick inserted in the center comes out clean.