Make the Cake:
Preheat oven to 350°F (175°C). Line a 10×15-inch jelly roll pan with parchment paper and lightly grease.
In a large bowl, beat eggs and sugar together until light and fluffy. Add vanilla. In a separate bowl, whisk together flour, cocoa powder, baking powder, and salt. Gently fold the dry ingredients into the egg mixture until fully combined.
Spread batter evenly into the prepared pan and bake for 12–15 minutes, or until the cake springs back when touched.

Roll the Cake:
While the cake is still warm, carefully invert it onto a clean kitchen towel dusted with powdered sugar. Peel off the parchment paper. Starting at the short end, roll the cake up with the towel. Let it cool completely rolled up.

Prepare the Filling:
In a medium saucepan, combine brown sugar, egg yolks, evaporated milk, and butter. Cook over medium heat, stirring constantly, until thickened (about 10 minutes). Remove from heat, then stir in toasted pecans, coconut, and vanilla. Let cool.

Assemble the Cake:
Carefully unroll the cooled cake. Spread the filling evenly over the surface, then gently roll the cake back up without the towel. Place seam-side down on a serving platter and refrigerate to firm up.

Make the Ganache:
Heat the heavy cream in a small saucepan until just beginning to simmer. Pour over chopped chocolate and butter in a bowl. Let sit 1 minute, then stir until smooth. Let cool slightly, then pour over the cake roll.

Decorate:
Sprinkle the top with toasted shredded coconut. Chill briefly to set the ganache before slicing.