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Position a rack in the center of the oven and preheat to 350°F. Lightly grease 3 (8-inch) round cake pans with sunflower oil and line each with a round of parchment paper cut to fit.
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Make the cake: In a large bowl, whisk together 1-1/4 cups water with the oil, egg yolks, and vanilla, then whisk in the 3/4 cup sugar.
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In a bowl, sift together the three flours with the baking powder and salt, adding back any bits left behind in the sifter. Whisk the flour mixture into the water mixture until very smooth; it will thicken up as it stands.
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In the clean, dry bowl of a stand mixer fitted with the whip attachment, whip the egg whites with the cream of tartar on medium-high speed until soft peaks form. Slowly add the 3 tablespoons sugar and whip until the whites are glossy and just hold a firm peak when lifted from the bowl, 1 to 3 minutes longer. Fold one-third of the whites into the batter, then gently fold in the remaining whites until just combined and no streaks remain. Divide the batter evenly among the prepared pans and smooth the tops. Bake until the cakes are golden, spring back to the touch, and a toothpick inserted near the center comes out clean, 25 to 30 minutes. Remove to a rack and let cool completely in the pan, about 1 hour. Loosen the edges with a knife or offset spatula and invert the cakes onto a rack or board. Peel away the parchment paper.
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Make the cream: Without tilting or shaking the cans, remove the coconut milk from the refrigerator and open the cans. Scoop out the firm coconut cream on the top, leaving behind the watery liquid in the bottom, and place in the bowl of a stand mixer fitted with the whip attachment. Whip on high until smooth and fluffy, about 1 minute, then beat in the powdered sugar, vanilla, and salt.
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To assemble, in a large bowl, toss the berries with the sugar and let stand for a few minutes. Place one cake layer on a serving plate or platter. Spread with one-third of the whipped coconut cream in an even layer, then scatter one-fourth of the berries evenly over the cream. Top with a second layer of cake and repeat with the remaining ingredients, ending with a mound of berries on the top layer. Chill the cake until firm for easier cutting, 1 to 2 hours, then slice into wedges and serve.