Preheat the oven to 350 degrees F.
Lightly butter 6 (3/4-cup) ramekins with 1 tablespoon of the butter. Coat the interior of the ramekins with the 2 tablespoons of sugar. Roll the sugar around each ramekin to ensure that all surfaces are evenly coated. Place the ramekins in a large baking dish or roasting pan and set aside.
In a mixing bowl, cream together 3/4 cup of the sugar and remaining 2 tablespoons butter. Add the egg yolks, 1 at a time, beating well after each addition. Stir in the flour. Gradually add the key lime juice; don't worry if the mixture looks curdled or appears to be separating. Season with the salt and add the milk and coconut and stir to combine.
In a separate bowl beat the egg whites just until they hold stiff peaks. Gently fold the egg whites into the yolk mixture in 3 additions. Do not overmix; the mixture will appear thin.
Divide the pudding cake mixture evenly among the ramekins and add enough hot water to the baking dish to come half way up the sides of the ramekins. Cover the ramekins with a sheet of parchment paper that has been liberally greased with butter or non-stick baking spray. Bake until the pudding cakes are puffed and lightly firm to the touch, 30 to 35 minutes. Remove from the oven and transfer to a wire cooling rack to cool slightly before serving.
To prepare the sauce: Puree the guava with the remaining 1/4 cup sugar and 1 tablespoon of lime juice for 2 minutes or until smooth. Strain through a fine-mesh sieve.
To serve, invert a ramekin with the warm pudding cake onto the center of a dessert plate. While clasping the sides of the plate with your fingers and using your thumbs to hold the base of the ramekin in place, give a couple of firm jolts downwards to release the cake from the ramekin. Place the plate on a level surface and slowly remove the ramekin. Spoon some of the guava sauce around the pudding cake, and place a mint sprig in the center of each cake. Serve while still warm