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To make the cake: Preheat the oven to 350°F. Grease and flour two 9" cake pans, or line with parchment circles and grease the parchment.
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In a large mixing bowl, beat the butter with the sugars, baking powder, ginger, vanilla, and salt until smooth. Scrape down the bowl, and beat for 30 seconds more. Beat in the first 2 eggs, one at a time.
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Add 2 tablespoons of the flour to the butter/egg mixture, mix to combine, then add the third and fourth eggs, beating well after each and scraping the bowl between additions.
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Add the remaining flour in three additions, alternating with the milk and ending with the last of the flour. Scrape the bowl once more, then mix in the rum and rum extract.
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Divide the batter between the prepared pans, level the tops, and bake for 30 to 32 minutes, until the top springs back when lightly touched in the center and the cake just begins to pull away from the edges of the pan. Remove from the oven and cool in their pans on a rack.
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To make the soak: Place the ginger, sugar, ginger beer or water, and rum in a medium saucepan. Bring to a boil, stirring occasionally until the sugar dissolves. Remove from the heat and set aside.
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To make the frosting: Combine the sugar, salt, and 1/2 cup of the ginger beer in a medium saucepan, and bring to a boil, stirring occasionally until the sugar dissolves. Boil for 3 to 5 minutes without further stirring, until the syrup reaches 240°F, then pour the hot syrup into the bowl of a stand mixer fitted with a whisk. Add the remaining ginger beer, rum extract, and rum.
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Sprinkle the meringue powder over the hot liquid, and whisk slowly until the powder dissolves.
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Increase the mixer's speed to medium and beat until the meringue powder mixture begins to get foamy, then turns white and increases in volume. Turn the mixer's speed to high, and beat for 3 to 4 minutes, until the meringue is thick and glossy looking.
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Add the butter, a few tablespoons at a time, with the mixer running. The mixture will thin out and look a little curdled for awhile; don't let that throw you. Keep the mixer going and add the rest of the butter. If you want more frosting or are planning to serve the cake in a warm environment, beat in the additional 1/2 cup shortening or butter. The frosting will transform itself into a smooth, silky, fluffy mass as you go.
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If you wish to create an ombré effect on the cake that mimics the cocktail, take 1 cup of the frosting from the bowl and mix in the molasses. Set aside.
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To assemble: Turn the cake layers out of their pans and place them right side up. If the layers have a dome on the top, use a long, serrated knife to trim them level. Brush the trimmed side with 2 to 3 tablespoons of the Dark and Stormy soak, then flip one of the layers over, trimmed side down, on a serving plate.
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Brush the other (bottom) side of the layer with more of the soak. Once it's absorbed, spread a 3/8" thick layer of frosting over it. Repeat the process with the second layer, placing it over the first.
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Spread a very thin coat of frosting over the top and on the sides of the cake; it's OK if this looks messy. This is called the crumb coat. Refrigerate the cake for 30 minutes to allow the crumb coat to firm up.
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After you can touch the crumb coat without leaving a fingerprint, use the rest of the frosting to put a finish layer on the top and sides of the cake. A Dark and Stormy cocktail is darker on top and lighter on the bottom, so if you like, use the tinted frosting around the top of the cake, blending it in as you move toward the base. Garnish with lime slices, if desired. Keep the cake chilled until 30 minutes before you wish to serve it.