For the choux balls: 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 evenly-sized (about 1 inch/2.5 cm) balls 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 the 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 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 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.
For assembling: Put the pastry cream in a piping bag with a sharp tip. Poke the tip into the choux balls and fill with pastry cream. Put the sugar in a saucepan with 1/2 cup water and dissolve. Boil to caramel (340 degrees F). Dip the choux balls into the caramel and arrange in a circle on a platter or cake stand. Continue dipping and build the balls up like a stone wall, preferably around an oiled cone (or flowerpot) to make a big tree. Spin some of the extra caramel around the tree. Set aside until ready to serve.