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1
Chill bowl of stand mixer and whisk attachment or large metal bowl and beaters for at least 15 minutes.
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2
In small saucepan over moderate heat, bring 1 cup cream to simmer. Transfer white chocolate to medium heatproof bowl, pour hot cream over, and whisk until smooth. Whisk in rum and salt. Let cool at room temperature until thickened slightly, about 1 hour.
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3
In chilled bowl of electric mixer fitted with whisk attachment, beat remaining 2 cups cream at moderately high speed until whisk leaves marks but cream does not quite hold soft peaks, 6 to 8 minutes. Turn mixer off, then add white chocolate mixture and beat just until stiff peaks begin to form, about 5 minutes. (Do not overbeat, or cream will curdle.) Refrigerate until firm, about 3 hours. (Frosting can be made ahead and refrigerated, covered, up to 8 hours.)
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4
Preheat oven to 350F. Line bottoms of cake pans with parchment paper.
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5
In large bowl, whisk together almond and coconut flours.
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6
In bowl of electric mixer fitted with whisk attachment, beat egg yolks at high speed until pale yellow and fluffy, 2 to 3 minutes. Reduce speed to moderately low and beat in rum and all but 1 tablespoon confectioner's sugar. Scrape down bowl, then increase speed to high and beat until pale and thick, about 1 minute. Reduce speed to low and gradually add almond and coconut flour mixture, scraping down bowl and folding in last of flour by hand. Set aside.
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7
In clean dry bowl of electric mixer fitted with whisk attachment, beat egg whites on moderate speed until very foamy, about 1 minute. Beat in cream of tartar, salt, and remaining 1 tablespoon confectioner's sugar. Increase speed to moderately high and beat until whites hold stiff peaks, about 2 minutes. Fold 1 cup beaten egg whites into yolk mixture to lighten, then gently fold in remaining whites. Divide batter between pans, smoothing tops, and bake until layers are golden brown and toothpick inserted in center comes out clean, 20 to 25 minutes. Invert pans onto wire racks and cool completely, at least 1 hour. Run knife around cake layers to loosen, invert onto rack, and peel off the parchment.
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8
Layer cake, using 1 cup frosting between each layer, then frost top and sides with remaining frosting. Press coconut flakes onto the frosting. Refrigerate until frosting is firm, at least 3 hours or overnight.
