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1
Heat the oven to 325 degrees F. Line a 15-by-10-inch jelly-roll pan with parchment paper and lightly coat the parchment with cooking spray.
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2
Melt half the butter (1 stick) in a small saucepan over low heat.
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3
Continue cooking until the butter is golden brown and smells toasty, occasionally scraping the bottom of the pan with a wooden spoon.
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4
Set aside to cool.
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5
In the bowl of a stand mixer fitted with the paddle attachment, cream the remaining butter with 1/2 cup of the sugar until the mixture is fluffy and pale.
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6
Add 1 egg and the teaspoon of rum.
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7
In a separate bowl, sift together the flour, baking powder, and salt.
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8
Add the dry ingredients to the butter mixture in three additions, alternating with the buttermilk and beating well after each addition.
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9
In a separate mixing bowl, combine the remaining 3 eggs and 1/4 cup sugar.
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10
Whip for 2 minutes on medium speed, then increase speed to high and whip until the mixture is thick and pale, 2 minutes more.
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11
Fold the whipped egg mixture into the cake batter, then fold in the browned butter until just combined.
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12
Pour the mixture into the prepared jelly-roll pan and spread to an even layer.
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13
Bake until cake is golden and a toothpick inserted into the center comes out clean, 30 to 35 minutes.
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14
Cool completely in the pan on a wire rack.
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15
Stir together the lime juice and sweetened condensed milk in a large bowl.
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16
In a separate bowl, whip the cream to soft peaks.
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17
Gently fold the whipped cream into the lime mixture.
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18
Cut the dried fruits into thin strips or cubes and toss with the vanilla and coconut extracts and lime juice.
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19
Turn cake layer out of pan onto a cutting board.
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20
Trim off the edges and cut the cake crosswise into three equal rectangles.
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21
Drizzle a few drops of rum over each strip of cake.
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22
Spread about 1/4 of the key lime cream evenly over each surface of two strips of the cake.
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23
Stack one frosted strip on top of the other.
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24
Place the remaining cake layer on top of the first two and frost the sides and top of the cake with the remaining cream.
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25
Decorate the top of the cake with the tropical fruit pieces.
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26
Serve at room temperature.
