Preheat oven to 400˚F. Line a baking sheet with parchment paper, set aside.
To make the pâte â choux, place a medium size saucepan over low heat. Add butter, sugar, water and salt, stirring frequently. Turn heat up to a low boil, remove from head and add in flour. Stir vigorously until the mixture forms a thick paste and no flour pockets remain. Place over medium heat and cook for 1-2 minutes stirring constantly to prevent burning (a thin film might form on the bottom of your pan).
Place hot paste into the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed for 2 minutes until it becomes slightly cooler. Then add the eggs one at a time until incorporated. The mixture will look shiny and will stick to the sides of the bowl. Beat for an additional 2 minutes on medium. Scoop dough into a large piping bag fitted with a large tip (a ½” in diameter). Pipe 4″-5″ logs onto the prepared baking sheet, spacing each two inches apart to prevent spreading and baking into each other. Using a pastry brush, gently and lightly coat each log with the beaten egg wash.
Place in the oven and bake for 20 minutes. After 20 minutes, open the oven, rotate the pan and reduce the heat to 350˚F, bake for an additional 20 minutes. Reduce oven heat to 300˚F and bake for 20 more minutes to allow the pastry shells to dry out. That makes a total of 60 minutes or 1 hour at three separate temperatures. Remove from the oven and allow to cool for 20 minutes.
Meanwhile to make the pastry cream, in a large mixing bowl, combine egg yolks and sugar, whisking until combined. Pour milk into a medium saucepan over medium heat, bring to a low boil then add the flour. Gradually pour the hot milk into the egg/sugar mixture to temper it, whisking constantly to prevent the eggs from scrambling. Return the mixture to the saucepan set over medium heat, add vanilla bean paste and butter, cook the mixture until it becomes very thick. Pour through a mesh sieve and set over an ice bath to cool (whisk for about 5 minutes or so). Store in the refrigerator until needed.
When the pastry shells have cooled, poke three holes into their bottoms. Place pastry cream into a large piping bag fitted with a small round tip and fill each hole. Once the shells are filled, heat the heavy whipping cream for the glaze in the microwave for 30 seconds. Stir in semi-sweet chocolate and salt. Dip each pastry into the chocolate glaze swirling to remove any excess chocolate. Set on the baking sheet and place in the refrigerator to set for about 30 minutes.