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Preheat oven to 325 degrees. Spray an 8-inch-square baking pan with nonstick cooking spray. Line pan with two pieces of parchment paper, creating a 1-inch overhang on all sides of the pan. Spray parchment paper with cooking spray.
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Make the Apple Topping: In a medium skillet, melt butter over medium-high heat. Continue cooking until butter becomes golden brown and fragrant. Reduce heat to medium and add sugar, 1/4 cup water, and apples. Cook until apples have softened slightly and liquid is thick and syrupy, about 5 minutes. Remove from heat; stir in Cognac, working carefully as alcohol may ignite. Return to heat, and cook until alcohol evaporates, about 30 seconds. Remove from heat and cool to room temperature.
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Make the Crumb Topping: In a medium bowl, whisk together flour, both sugars, cinnamon, and salt. Add melted butter and, using your fingers, mix together to form large crumbs; set aside.
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Make the Cake: In a large bowl, whisk together flour, baking powder, baking soda, and salt; set aside. In the bowl of an electric mixer fitted with the paddle attachment, cream together butter and sugar until light and fluffy, 2 to 3 minutes. Add creme fraiche, egg, yolk, and Cognac; mix until well combined. Add flour mixture and mix until well combined.
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Pour cake mixture into prepared baking pan. Top with apples and their liquid to cover; sprinkle evenly with crumb topping. Transfer to oven and bake until a tester inserted into the center of cake comes out clean, about 55 minutes. Remove from oven and let cool completely before serving.