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For the cake, grease a 9" x 13" cake pan; line the bottom with parchment paper.
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Set aside.
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In a large bowl, beat the butter with the brown sugar until fluffy; beat in the eggs, one at a time.
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Beat in the vanilla.
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In a separate bowl, whisk together the flour, baking powder, salt, baking soda, cinnamon, ginger, allspice and nutmeg.
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Stir into the butter mixture alternately with the applesauce and buttermilk, making 3 additions of dry ingredients and 2 of wet.
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Spread in the prepared pan.
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Bake in the centre of a 350 degree F oven until the cake tester inserted in the centre comes out clean, about 35 minutes.
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Let cool in the pan on a rack.
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For the icing, melt the butter in a saucepan over medium heat.
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Add the brown sugar and corn syrup; bring to a boil, stirring constantly.
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Boil just until blended, about 1 minute.
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Remove from the heat.
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Using a wooden spoon, stir the milk and vanilla into the butter mixture.
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Again using a wooden spoon, beat in the icing sugar until smooth and spreadable; let cool for 1 minute.
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Spread over the top of the cake just to the edge, letting the icing drizzle over slightly.