Line a quarter-sheet baking pan with plastic wrap, making sure the sides of the pans are also covered. Set aside.
Pour 1 ½ cups milk into a large heavy bottomed sauce pot. Add ½ cup of sugar. Whisk together. Place over medium-high heat. Stir occasionally to make sure the milk does not burn along the sides or bottom of the pan. Bring to a boil.
Meanwhile, in a medium bowl, whisk together cornstarch, cocoa powder, and remaining ¼ cup sugar. Add in remaining ½ cup milk and egg yolks. Whisk until smooth.
Once milk has boiled, reduce heat. Slowly add a little bit of the milk into the cornstarch mixture. Whisk constantly to distribute the heat. Continue to add milk and whisk until only a couple of tablespoons of milk are left in the saucepan. Pour all the egg-milk mixture back into the saucepan.
Whisk constantly over medium-low heat until the custard has thickened. Keep whisking and let the custard come to boil to cook out the cornstarch flavor. Remove from heat. Whisk in butter until smooth. Add salt and vanilla and whisk until combined.
Pour the hot pastry cream into the prepared plastic wrapped sheet pan. Spread evenly. Cover with plastic wrap and freeze for 10-15 minutes until the pastry cream has cooled to room temperature. Remove from freezer and transfer to an airtight container. Place plastic wrap directly on pastry cream to avoid condensation. Let chill in the refrigerator until ready to use.