Preheat the oven to 350 degrees F. In a small saucepan, heat the milk and 2 tablespoons butter together. Using an electric mixer fitted with a wire whip, combine the eggs and sugar together. Beat on medium-high speed for about 8 minutes, or until the mixture is pale yellow, thick and has trippled in volume. With the machine running slowly add the heated milk. In a mixing bowl sift the flour, baking powder, and salt together. Fold the flour mixture into the egg mixture and mix thoroughly so that there are no lumps and the mixture is smooth. Fold in the vanilla. Grease a 1/2 sheet pan with 1 tablespoon butter. Sprinkle with a tablespoon of sugar. Pour cake batter evenly into pan and bake for about 15 to 20 minutes or until the edges are golden and the cake springs back when touched. Cool for 2 minutes. Using a thin knife, loosen edges and flip out onto a piece of parchment paper. Peel the parchment paper from the cake. Replace with a new sheet of parchment and flip back into the baking sheet.
In a heavy bottom saucepan, over medium heat, combine the evaporated milk, sugar, and salt together. Bring the liquid to a boil. Temper the egg yolks into the hot cream. Stir in the butter. Reduce the heat to a simmer and cook for 12 minutes, stirring constantly.
Remove the pan from the heat and stir in the pecans, coconut and vanilla. Cool the mixture completely. Set the mixture aside. In a non-stick saucepan, over medium heat, combine the cream and chocolate pieces. Stir until the chocolate has melted completely and the mixture is smooth. Remove the pan from the heat and stir to cool, lifting the mixture out of the pan several times with a spoon. The mixture should be glossy and slightly thick. Set the mixture aside.
To assemble, spread the coconut/pecan icing evenly over the sheet cake. Starting from one of the long sides, roll the cake up tightly, like a jelly roll. Wrap the cake tightly in parchment paper and then wrap the cake in plastic wrap. Place the cake in the refrigerator and chill for 1 hour. Remove the cake from the refrigerator and carefully unwrap the cake. Place a wire rack over a piece of parchment paper. Place the cake, seam side down, on the rack. Evenly pour the chocolate ganache over the cake, letting the chocolate run down the sides of the cake. If the poured chocolate does not ice the cake completely, use a clean knife and spread the chocolate evenly over the unexposed areas. Using a fork, run the tines of the fork back and forth across the cake. Refrigerate the cake for 2 hours or until the ganache is set. Remove the cake from the refrigerator and using a long spatula, carefully remove the cake from the rack and place on a serving platter. Garnish the cake with fresh mint sprigs and powdered sugar. 12 servings