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For this recipe, we perfer to use a 9-inch pan with the sides that are at least 2-inches high. Alternatively, a 1-inch safe oven-safe skillet (either cast iron or stainless steel) can be used not only to cook the pineapple, but to bake th cake as well. If using a skillet instead of a cake pan, cool the juices directly in the skillet while making the batter; it's OK if the skillet is still warm when the batter is added. For a cake with a light, tender crumb, it is important to cream the butter and sugar well and to have the eggs at room temperature.
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Lightly spray 9-inch round, 2-inch deep cake pan with nonstick cooking spray, set aside.
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FOR THE PINEAPPE TOPPING:
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combine pineapple, brown sugar, and orange zest in 10-inch skillet; cool over medium heat, stirring occasionally during first 5 minutes, until pineapple is translucent and has light brown hue, 15-18 minutes. Empty fruit and juices into mesh strainer or colander, set over medium bowl. Return juices to skillet, leaving pineapple in strainer, (you should have about 2 cups cooked fruit). Add orange juice to skillet and simmer juices over medium heat until thickened, beginning to darken, and mixture forms large bubbles, 6-8 minutes, adding any more juices released by fruit to skillet after 4 minutes. Off heat, whisk in butter and vanilla; pour caramel mixture into prepared cake pan. Set aside while preparing cake. (Pineapple will continue to release liquid as it sits: do not add this liquid to already-reduced juice mixture.).
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FOR THE CAKE:
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Adjust oven rack to lower-middle position and heat oven to 350u00b0F Whisk flour, baking powder, salt, and cardamom in medium bowl; set aside.
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In bowl of standing mixer fitted with flat beater, cream butter and sugar at medium-high speed until light and fluffy. 3-4 minutes. Reduce speed to medium, add vanilla, and beat to combine; one at a time, add whole eggs, the egg white, beating well, and scraping down bowl after each addition. Reduce speed to low; add about 1/3 of the flour mixture and beat until incorporated. Add half of milk and beat until incorporated; repeat, adding half of remaining flour mixture and remaining milk, and finish with remaining flour. Give final stir with rubber spatula, scraping bottom and sides of bowl to ensure that batter is combined. Batter will be thick.
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TO BAKE:
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Working quickly, distribute cooked pineapple in cake pan in even layer, gently pressing fruit into caramel. Using rubber spatula, drop mounds of batter over fruit, then spread batter over fruit and to sides of pan. Tap lightly against work surface to release any air bubbles. Bake until cake is golden brown and toothpick into center comes out clean, 45-50 minutes. Cool 10 minutes on wire rack, then place inverted serving platter over cake pan. Invert cake pan and platter together; life off cake pan. Cool to room temperature, about 2 hours. Then cut into pieces and serve.
