Preheat oven to 400 degrees F (205 degrees C) and place the rack in the center of the oven. Line a baking sheet with parchment paper. Draw an eight inch (20 cm) circle on the parchment paper to use as a guide when piping the choux pastry.
In a bowl sift together the flour, sugar and salt. Set aside.
Place the butter, milk, and water in a heavy saucepan over medium heat and bring to a boil. Remove from heat and, with a wooden spoon, quickly stir in the flour mixture. Return to heat and stir constantly until the dough comes away from the sides of the pan and forms a thick smooth ball (about 3 to 4 minutes). Transfer the dough to your electric mixer, or with a hand mixer, and beat on low speed a minute or two to release the steam from the dough. Once the dough is lukewarm, start adding the lightly beaten eggs, in small additions, and continue to mix until you have a smooth thick shiny paste. (Note: To test to see if the dough is the right consistency; pinch off about a teaspoon of the dough. Then with your thumb and index finger slowly pull the dough apart. The dough should stretch, not break. If it breaks, beat in a little more egg.)
Transfer the dough to a large piping bag fitted with a large plain tip (#4). Pipe a 1 inch (2.54 cm) thick ring of dough just inside the 8 inch (20 cm) circle that is drawn on the parchment paper. Pipe another 1 inch (2.54 cm) thick ring inside the first ring, making sure the rings of dough are touching. Pipe a final ring of dough on top, along the center seam between the first two rings of dough. Then, with moistened fingertips, gently smooth the rings of dough. In a small cup, beat together the egg and salt for the glaze. With a pastry brush, gently brush the glaze on the top and sides of the dough. Sprinkle with the shaved almonds.
Bake for 15 minutes and then reduce the oven temperature to 350 degrees F (177 degrees C). Bake for a further 35 to 45 minutes or until the Paris Brest is a golden brown color, puffed, and crisp. Turn the oven off and, with the oven door slightly ajar, let the shell dry out for about 30 minutes. Remove from oven and finish cooling on a wire rack.
For whipped cream: In a large mixing bowl place the whipping cream, vanilla extract, and sugar and stir to combine. Cover and chill the bowl and whisk in the refrigerator for at least 30 minutes. When chilled, whip the mixture until soft peaks form.
To Assemble: Split the pastry shell in half horizontally and remove any wet pieces of dough. Then spoon (or pipe) the whipped cream into the hollow center of the shell. Place the top half of the pastry shell on the whipped cream and dust with powdered (confectioners or icing) sugar.
Makes one large 8 inch (20 cm) Paris Brest.