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Preheat the oven to 340°F (170°C). Thoroughly grease your small Bundt pan and dust it lightly with flour.
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Beat the eggs, egg yolk, sugar, and salt for a few minutes until pale, thick, and fluffy.
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Add the vanilla bean paste, buttermilk, and oil and mix briefly until combined.
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Mix the flour and baking powder together. Add to the wet ingredients and mix just until no dry streaks remain. Avoid overmixing.
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Transfer about ¼ of the batter to a separate bowl. Stir in the brown sugar, toasted hazelnuts, Ceylon cinnamon, and milk.
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Spoon half of the plain batter into the prepared Bundt pan. Add half of the cinnamon batter on top. Repeat with the remaining plain batter and cinnamon batter.
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Run a fork or thin knife through the batter with just a few swirling motions. Don’t mix too much or you’ll lose those distinct cinnamon layers.
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Bake for approximately 35–40 minutes, or until a toothpick inserted into the cake comes out clean or with just a few moist crumbs.
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Let the cake cool briefly in the pan before carefully turning it out onto a wire rack. Allow it to cool completely.
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For the glaze, whisk the powdered sugar with 1½ tablespoons heavy cream and a little vanilla bean paste. Add more cream as needed until you have a thick, pourable vanilla glaze. Drizzle over the cooled cake.