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Preheat the oven to 350 degrees Grease an 8 inch round pan and set aside.
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Whisk the flour, baking powder, and salt together in a small bowl.
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In a larger bowl, beat the eggs with a whisk until foamy, then add the sugar and whisk until smooth. Add the molasses, water, and both extracts. Whisk in half of the flour mixture, then half the melted butter, then the remaining flour and butter. Whisk until you have a smooth, thick batter.
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Fold in the apples and pour batter into prepared pan, spreading with a spatula or spoon until it's fairly even.
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Bake for 50-60 minutes, or until the top of the cake is golden brown and a knife inserted in the center comes out clean.
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Serve warm or at room temperature. If you want to remove the cake from the pan, wait until the cake is completely cooled.
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Serve with ice cream, whipped cream, or plain for breakfast.