1Heat the oven to 325°F. Line a 15-by-10-inch jelly-roll pan with parchment paper and lightly coat the parchment with cooking spray.
2Melt half the butter (1 stick) in a small saucepan over low heat. Continue cooking until the butter is golden brown and smells toasty, occasionally scraping the bottom of the pan with a wooden spoon. Set aside to cool.
3In 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. Add 1 egg and the teaspoon of rum. In a separate bowl, sift together the flour, baking powder, and salt. Add the dry ingredients to the butter mixture in three additions, alternating with the buttermilk and beating well after each addition.
4In a separate mixing bowl, combine the remaining 3 eggs and 1/4 cup sugar. Whip for 2 minutes on medium speed, then increase speed to high and whip until the mixture is thick and pale, 2 minutes more.
5Fold the whipped egg mixture into the cake batter, then fold in the browned butter until just combined. Pour the mixture into the prepared jelly-roll pan and spread to an even layer. Bake until cake is golden and a toothpick inserted into the center comes out clean, 30 to 35 minutes.
6Cool completely in the pan on a wire rack.
For the key lime cream:
1Stir together the lime juice and sweetened condensed milk in a large bowl. In a separate bowl, whip the cream to soft peaks. Gently fold the whipped cream into the lime mixture.
For the tropical fruits:
1Cut the dried fruits into thin strips or cubes and toss with the vanilla and coconut extracts and lime juice.
To assemble the cake:
1Turn cake layer out of pan onto a cutting board. Trim off the edges and cut the cake crosswise into three equal rectangles. Drizzle a few drops of rum over each strip of cake.
2Spread about 1/4 of the key lime cream evenly over each surface of two strips of the cake. Stack one frosted strip on top of the other. Place the remaining cake layer on top of the first two and frost the sides and top of the cake with the remaining cream. Decorate the top of the cake with the tropical fruit pieces. Serve at room temperature.
Beverage pairing:Kent Rasmussen Winery Late Harvest Gewürztraminer, Sonoma. The aromas of this sweet wine suggest the raisinated grapes from which it was made. Its caramel and toffee overtones will merge seamlessy with the rum, while the tropical fruits in the cake are matched perfectly by the character of super-ripe Gewürztraminer.