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Position rack in the lower third of the oven and preheat overn to 325°F. Coat a 12-cup Bundt pan with nonstick cooking spray and dust it lightly with flour.
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Sift together the flour, baking powder, and salt into a large bowl.
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In the bowl of a stand mixer fitted with the paddle attachment, beat butter and vanilla extract on medium speed until smooth and creamy. Add sugar and beat until evenly mixed, about 1 minute. Slowly pour in oil. Beat until light and fluffy, about 2 minutes. Add eggs and egg yolks, 1 at a time, beating thoroughly after each addition.
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Reduce mixer speed to low. Stir in flour mixture and buttermilk in three alternating additions, ending with the buttermilk, and continue to mix on low speed until the batter is smooth and no lumps remain. Turn off mixer and fold batter several times by hand to ensure everything is well incorporated, and then pour into prepared pan.
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Bake for 60 to 70 minutes, until the cake is golden, the top springs back when lightly pressed, and a cake tester inserted into the center comes out clean. Place pan on a wire rack and let cool completely, about 2 hours.