To make the frosting, whisk the egg yolks, milk, and vanilla in a large saucepan until blended. Add the sugar and butter and cook over medium heat for 12 minutes, stirring constantly, until the mixture is thick and golden brown. Remove from the heat and mix in the coconut and pecans. Allow the frosting to cool to room temperature.
Make the Cake: Line a 15 x 10 1-inch jelly roll pan with parchment paper.
In a mixing bowl, beat the baking powder; salt, eggs, and sugar until it forms a thick lemon-colored batter.
In a separate bowl, combine the flour and cocoa and fold into the egg mixture. Stir in the vanilla.
Spread the batter in a jelly roll pan and bake at 400 degrees F for 12 minutes, or until the center of the dough springs back when touched. Remove from the oven and trim the crust from the edges.
Dust a towel or cloth with the powdered sugar and turn the cake out onto the towel. Peel off the parchment paper.
Spread the coconut-pecan filling onto one side of the cake, roll the cake, and cover it with the towel. Place on a wire rack, seam side down, and allow it to cool.
Remove the cloth and transfer the roll to a serving plate, seam side down. Dust the roll with cocoa and slice diagonally to serve.