Preheat the oven to 200°C and line two baking sheets with parchment paper.
Combine water, milk, butter, sugar, and salt in a medium saucepan and bring to a rolling boil over medium heat.
Remove from heat, dump in sifted flour at once, and stir vigorously with a wooden spoon until no dry streaks remain.
Return the pan to medium heat and cook the dough, stirring, until it forms a smooth mass and leaves a thin film on the pan, about 2 minutes.
Transfer the dough to a large bowl and let cool 5 minutes until warm but not hot to the touch.
Beat in the eggs one at a time with a mixer, fully incorporating each before adding the next, until the dough is glossy and forms a thick V when lifted.
Fill a piping bag fitted with a large round tip and pipe 12 logs about 12 cm long on the prepared sheets, spacing well.
Smooth any peaks with a damp finger and mist lightly with water.
Bake at 200°C for 15 minutes, then reduce to 175°C and bake 15–20 minutes more until deeply golden and puffed.
Turn off the oven, poke a small slit in each éclair with a knife, and return to the oven with the door ajar for 10 minutes to dry.
Cool shells completely on a rack.
For the pastry cream, heat milk in a saucepan until steaming but not boiling.
In a bowl, whisk sugar, cornstarch, egg yolks, and salt until thick and pale.
Slowly whisk in half the hot milk to temper, then return the mixture to the saucepan with the remaining milk.
Cook over medium heat, whisking constantly, until thick bubbles form and the cream holds lines, about 2–3 minutes after boiling.
Off heat, whisk in butter, pistachio paste, and vanilla until smooth.
Strain through a sieve into a clean bowl, press plastic wrap onto the surface, and chill until cold, about 1 hour.
Whip the cold heavy cream to soft peaks and fold into the chilled pistachio pastry cream until smooth.
Fit a piping bag with a small round or Bismarck tip and fill with pistachio cream.
Pierce the ends or underside of each éclair in two spots and pipe cream until the shells feel heavy.
For the glaze, heat 40 ml cream until steaming, pour over chopped chocolate, let sit 1 minute, then stir until glossy.
Dip the tops of filled éclairs into the chocolate glaze, letting excess drip back into the bowl.
Sprinkle with chopped pistachios and chill 20–30 minutes to set before serving.