Heat the oven to 375°F/190ºC. Put the water, salt, butter, and sugar in a roomy saucepan, 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 non-stick 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.