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Preheat oven to 350°F.
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Butter a 9-inch round springform pan and line the bottom with a round of parchment paper.
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Butter and flour the parchment paper.
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Combine the eggs, oil, milk, sugar, and vanilla extract in a mixing bowl and mix well.
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Sift in enough flour to make a batter with the consistency of heavy cream.
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Pour the batter into the cake pan.
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Arrange the apple slices on top of the batter in concentric circles and sprinkle with the cinnamon.
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Bake until the batter has set, about 20 minutes.
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For the glaze, mix together the butter, sugar and egg.
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Pour glaze evenly over the pudding and bake until a sharp knife inserted into the center comes out clean, 15 to 20 minutes longer.
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Let cool slightly before removing the pudding from the pan.
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Serve while still warm.