Place the butter, salt, sugar and water in a saucepan and heat this on medium heat until the butter is melted and the water just starts to boil.
Immediately add all the flour to the water (remove the saucepan from the heat for this), and with a wooden spoon quickly mix the flour into the water until all the water has been absorbed. Keep mixing until the dough comes off easily from the side and the bottom of the saucepan when mixed.
Transfer the dough into a bowl.
Using a hand mixer, beat in the eggs into the dough one at a time. The dough should become thick and glossy.
Finally add the vanilla and mix it in.
Line a baking tray with parchment paper.
Transfer the dough into two pastry bags - one with a large round tip (I used Wilton 2A) and another with a star tip (I used Wilton 195). See note 1 if you do not have these tips
Using the round tipped bag, pipe a 5-6 inch diameter circle on the parchment paper. Remember to leave room between the circles as they do expand.
Right on top of the piped circle, pipe another layer of the choux pastry with the round tip.
On top of this, pipe a third layer of choux pastry with the star tip.
Brush the top with beaten egg mix.
Place the tray in the preheated oven and let it bake for about 10 minutes.
Lower the temperature to 350°F and bake for another 15-20 minutes, until the choux pastry is nice and puffed up, golden brown on top and crisp to your touch.
Remove from the oven and prick each pastry with a toothpick to let out the air.
Let it cool to room temperature.
Assembly
With a serrated knife, cut the Paris Brest pastry in half carefully.
Pipe the salted chocolate filling (recipe below) in the middle and place the top back on.
Drizzle caramel (recipe below) carefully (this is going to be hot, so take care to not burn yourself!).
The caramel will harden in a few minutes.
Notes
Note 1 - If you don't have star tips, you can pipe the 3 circles with a round tip or use a ziploc bag that has been cut at the bottom, and then use a fork to create the grooves lightly on the top circle.