For the choux pastry: Preheat the oven to 375 degrees F/190 degrees C. Put 1 1/2 cups/375 ml water, the butter, sugar and salt in a roomy saucepan and bring to a boil. Remove from the heat and add the flour all at once, beating until it forms a ball and pulls away from the sides of the pan. Return the pan to the heat for a minute or two and beat to dry it out a little. (Removing moisture will allow the paste to absorb more egg, which will ultimately make the puffs lighter.)
Remove the pan from the heat. Beat in the eggs, the equivalent of one at a time, until fully incorporated. (Do not try to add them at once because they can't be incorporated quickly enough, it makes a total mess, and furthermore you may not need all of them.) After the third addition of egg, add only enough to make dough that will fall, glossy and heavy, from the spoon. Beat in the vanilla. Spoon the dough into a piping bag.
Mix a bit of milk into any remaining egg (if no egg left, use only milk). Have at the ready with a pastry brush for glazing. Pipe the dough into finger-like logs on a nonstick baking sheet, leaving a good 2 inches/5 cm between them so they have room to expand. Brush the tops with the glaze and press any snouts down with a fingertip. Bake until puffed up, light, dry and golden, 25 to 30 minutes. Cool on a wire rack, so they can breathe all around and will keep their shape and not go soggy.
For the coffee pastry cream: Put the milk in a saucepan. Scrape the vanilla seeds into the milk and throw in the pod. Bring just to a boil, remove from the heat, cover and set aside to infuse for 10 to 15 minutes.
Beat the yolks and sugar to pale, thick ribbons. Gradually beat in the flour and salt. Whisk the hot milk into the egg mixture in a thin stream. Return the mixture to the saucepan and add the coffee flavoring. Bring to a boil over medium heat, stirring constantly until the mixture thickens. Strain and set aside to cool with a buttered piece of waxed paper on top to prevent a skin from forming. When cool, fold in the whipped cream and transfer to a piping bag with a small tip (I find that one with a jagged edge works best).
For the coffee glaze: Put the confectioners' sugar in a bowl. Add 1/4 cup water and the coffee essence and stir to make a thin icing. Add more water if necessary.
Slice the eclairs in half lengthwise. Spread the bottom generously with pastry cream, as if you're making a fat sandwich. Spread the glaze on top of the top piece and lay it on the cream.