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Preheat the oven to 350 F.
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Toss the prepped apples with the lemon juice and 2 tablespoons of the brown sugar. Set aside.
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Prepare a 9x2 inch cake pan by greasing the bottom of the pan with a little oil or shortening and then layering a cut out circle of parchment paper on it.
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In a small, heavy saucepan, melt the 4 tablespoons of butter. Using a pastry brush, transfer about 1 tablespoon of the butter to the cake pan, brushing the sides of the pan and the bottom over the parchment paper.
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Add the remaining brown sugar to the saucepan, and then also add all of the juices that have drained out of the apples. Stir constantly and bring to a boil--when it boils, stop stirring. Simmer for 3 minutes, until it is bubbling and a deep amber in color.
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Pour this caramel mixture over the bottom of the cake pan--do not scrape the saucepan. Tilt the cake pan to evenly coat the bottom of the cake pan with the caramel sauce. Ignore any hardening--baking will re-melt it.
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Beginning in the center of the cake pan, place the apple slices in an overlapping spiral, like this:
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Set the pan aside, and make the cake batter.
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Whisk together the cake flour, sugar, baking powder, baking soda, cinnamon and salt in a stand mixer bowl.
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Fit the bowl into the stand mixer with the paddle attached.
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Whisk together the egg yolks, sour cream and vanilla. Set aside.
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Add the 9 tablespoons room temperature butter to the mixer bowl. Begin mixing it into the flour mixture on low speed. When the dry ingredients become moistened, beat on medium speed for a minute.
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Add 1/3 of the liquid mixture, and beat on medium speed for 30 seconds. Scrape down the sides and bottom of the bowl, and repeat twice more, adding all of the liquid ingredients.
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Scrape the cake batter onto the prepared apples in blobs across the cake pan--do not pour all in one pile, because the batter is thick and you do not want to disrupt the apples. Use a small offset spatula to smooth the batter evenly over the apples.
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Bake for 35-45 minutes, until the cake is golden brown, a cake tester inserted into the center of the cake comes out clean, and the cake springs back when pressed lightly in the center.
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While the cake is baking, toast and chop the pecans.
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When you remove the cake from the oven, immediately use a small metal spatula to run between the cake and the sides of the pan to loosen the cake. Invert the cake at once onto a serving plate-leaving the pan in place for 1-2 minutes before lifting it off of the cake.
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If any apples pieces stick, just quickly add them back to the cake.
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Sprinkle the chopped toasted pecans over the cake evenly. Serve warm or at room temperature.