Preheat oven to 200°C and line a baking sheet with parchment paper.
In a medium saucepan combine water, milk, butter, sugar, and salt, bring to a boil over medium heat, then remove from heat and stir in flour until a dough forms.
Return pan to medium heat and cook the dough, stirring constantly, until it pulls from the sides and leaves a thin film on the pan, about 2 minutes.
Transfer dough to a mixing bowl and beat for 1 minute to cool slightly.
Add eggs one at a time, beating well after each addition until the batter is smooth, glossy, and falls from the spoon in a thick V-shape.
Spoon the dough into a piping bag fitted with a 1/2-inch round tip and pipe 12 logs about 12 cm long onto the baking sheet, spacing well.
Smooth any peaks with a damp fingertip and mist lightly with water.
Bake at 200°C for 10 minutes, then reduce to 180°C and bake 20–25 minutes more until puffed, deep golden, and hollow sounding.
Turn off the oven, poke a small hole in each eclair with a skewer, and return to the oven with the door ajar for 10 minutes to dry.
Cool shells completely on a wire rack.
For pastry cream, heat milk with half the sugar in a saucepan until steaming.
In a bowl whisk remaining sugar, cornstarch, and egg yolks until pale, then slowly whisk in hot milk.
Return mixture to the saucepan and cook, whisking constantly, until thick bubbles appear and cream is glossy and thick.
Remove from heat and whisk in butter and vanilla, then press plastic wrap directly on the surface and chill until cold.
Whip cold heavy cream to soft peaks and fold into chilled pastry cream to make light crème diplomat.
Poke two small holes on the underside of each shell and pipe the cream into the eclairs until filled and slightly heavy.
For caramel glaze, combine sugar and water in a clean saucepan and cook without stirring until a deep amber color.
Carefully whisk in warmed cream, then butter and salt, cooking briefly until smooth and slightly thickened.
Pour caramel into a shallow heatproof bowl and let cool until thick but still fluid.
Dip the tops of filled eclairs into the caramel, letting excess drip off, and set on a rack to set.
Chill eclairs 30 minutes to set the caramel and firm the filling before serving.