Make the cupcakes: Preheat the oven to 350 degrees F. Line a 12-cup muffin pan with liners. Whisk the flour, baking powder and salt in a medium bowl. Whisk the eggs, 3/4 cup granulated sugar, the brown sugar, coconut milk, vegetable oil and vanilla and coconut extracts in a large bowl. Add the flour mixture and whisk until smooth.
Divide the batter among the prepared muffin cups, filling each about three-quarters of the way. Bake until the tops of the cupcakes are golden and spring back when gently pressed, 18 to 20 minutes.
Meanwhile, combine the rum, lime juice and the remaining 1 tablespoon granulated sugar in a small saucepan over medium heat. Bring to a simmer and cook, stirring, until the sugar dissolves. When the cupcakes are done, poke them all over with a toothpick and brush with the rum syrup while still warm. Let cool 5 minutes in the pan, then transfer to a rack to cool completely.
Meanwhile, make the frosting: Bring the pineapple juice to a boil in a small saucepan. Reduce the heat to a simmer and cook until syrupy and reduced to 2 to 3 tablespoons, 6 to 8 minutes. Transfer to a small bowl and refrigerate until completely cool. Beat the butter in a large bowl with a mixer on medium-high speed until creamy, about 1 minute. Reduce the mixer speed to low and beat in the confectioners' sugar and salt. Increase the speed to medium high and beat until light and fluffy, about 3 minutes. Add the vanilla and coconut extracts and beat 1 minute, then add the rum and the reduced pineapple juice and beat to combine. Spread the frosting on the cupcakes and top with the shredded coconut. Decorate with maraschino cherries and cocktail umbrellas.
Photograph by Ryan Dausch