Prepare the Cake
Preheat the oven to 400°F (204°C).
Grease a 15×10-inch sheet pan and set aside.
In a large mixing bowl, whisk together the flour, sugar, and cocoa powder.
In a medium saucepan over medium heat, combine the water, butter, shortening, and German baking chocolate.
Bring the mixture to a gentle boil, stirring until smooth and fully melted.
Pour the hot chocolate mixture into the dry ingredients and whisk until combined.
Stir in the buttermilk.
Add the baking soda and salt, mixing well.
Add the eggs one at a time, stirring after each addition.
Mix in the vanilla extract until the batter is smooth.
Pour the batter into the prepared sheet pan and spread evenly.
Bake for 20–22 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool completely before frosting.
Prepare the Frosting
In a medium saucepan, combine the evaporated milk, egg yolks, butter, granulated sugar, and brown sugar.
Cook over medium heat until the mixture comes to a gentle boil.
Reduce the heat to low and simmer for 10–12 minutes, stirring frequently, until thickened.
Remove from heat.
Stir in the shredded coconut, chopped pecans, and vanilla extract.
Allow the frosting to cool slightly until spreadable.
Assemble
Spread the frosting evenly over the cooled cake.
Refrigerate for 1–2 hours to allow the frosting to set.
Slice and serve.