Heat the butter, water, milk, salt, and sugar together in a saucepan over medium heat. Stir to dissolve, and bring to a simmering point.
As soon as the mixture starts to boil, remove it from the heat, and add the sifted flour in one go. Whisk vigorously with a Rubber spatula until it forms one smooth ball and flour bits are not visible. Place the saucepan back on the stove and cook the dough for about 3 minutes over medium heat, mixing constantly. It is ready when a skin forms on the bottom of the saucepan.
Transfer the dough to a bowl or stand mixer fitted with the paddle attachment and let it cool slightly before starting to mix the egg in. Only add a small amount of egg at a time, consistently whisking after each addition to let it incorporate. Stop when the mixture is glossy and silky and forms a V when slowly dropping off the whisk.
Transfer the mixture to a piping bag fitted with a round nozzle, pipe 20 equal-sized buns on the baking mat, and place the tray into the freezer for a minimum of 10 minutes.
Take the craquelin sheet out from the freezer, peel off the mats, and cut out 20 circles with a cookie or pastry cutter. They need to be slightly larger in diameter than the choux buns.
Take the choux buns out of the freezer and place one craquelin disk on top of each one. The craquelin disk should be frozen and very firm at this point; work quickly to avoid it melting.
Put the tray of choux au craquelin back into the freezer for 15 minutes.
Pre-heat oven to 200C/392F
Lower the oven temperature to 170C/338F and bake the choux au craquelin for 40 minutes. Do not open the oven door in the first 30 minutes. They are ready when golden brown in color, nicely puffed up, and do not deflate at room temperature.
Cool the choux au craquelin on the perforated air mat or, if baked on parchment paper, cool on a wire rack before filling or storing them.