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1
Make cake layers:
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Preheat oven to 350u00b0F and oil 3 9 inch cake pans. Line bottoms of pans with rounds of parchment or wax paper. Sift together sugar, flour, cocoa powder, baking powder, baking soda, and salt into a large bowl. Whisk together whole milk, butter, whole egg, yolk, vanilla, and almond extract in another large bowl until just combined. Beat egg mixture into flour mixture with an electric mixer on low speed, then beat on high speed 1 minute. Reduce speed to low and beat in water until just combined (batter will be thin). Divide batter among cake pans (about 1 1/2 cups per pan) and bake in upper and lower thirds of oven, switching position of pans and rotating them 180 degrees halfway through baking, until a tester comes out clean, 20 to 25 minutes total.
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Cool layers in pans on racks 15 minutes. Run a thin knife around edges of pans and invert layers onto racks. Carefully remove parchment or wax paper and cool layers completely.
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Make filling:
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Reduce oven temperature to 325u00b0F
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Spread coconut in a large shallow baking pan and pecans in another. Bake pecans in upper third of oven and coconut in lower third, stirring occasionally, until golden, 12 to 18 minutes. Remove pans from oven.
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Increase oven temperature to 425u00b0F
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Pour condensed milk into a 9-inch deep-dish pie plate and cover tightly with foil. Bake milk in a water bath in middle of oven 45 minutes. Refill baking pan with water to reach halfway up pie plate and bake milk until thick and brown, about 45 minutes more. Remove pie plate from water bath.
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Stir in coconut, pecans, and vanilla and keep warm, covered with foil.
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Make glaze while milk is baking:
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Melt butter in a 3-quart saucepan. Remove pan from heat and add chocolate and corn syrup, whisking until chocolate is melted. Transfer 1 cup glaze to a bowl, reserving remaining glaze at room temperature in pan. Chill glaze in bowl, stirring occasionally, until thickened and spreadable, about 1 hour.
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Assemble cake:
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Put 1 cake layer on a rack set over a baking pan (to catch excess glaze). Drop half of coconut filling by spoonfuls evenly over layer and gently spread with a wet spatula. Top with another cake layer and spread with remaining filling in same manner. Top with remaining cake layer and spread chilled glaze evenly over top and side of cake. Heat reserved glaze in pan over low heat, stirring, until glossy and pourable, about 1 minute. Pour glaze evenly over top of cake, making sure it coats sides. Shake rack gently to smooth glaze.
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Chill cake until firm, about 1 hour. Transfer cake to a plate.
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Cooks' notes:
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Cake keeps, covered and chilled, 3 days. Bring to room temperature before serving.
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For easier handling when assembling cake, place bottom layer on a cardboard round or the removable bottom of a tart or cake pan.
