Pre-heat the oven to 400°F degrees. Line baking sheets with parchment and set aside.
Place the butter, milk, water, sugar, and salt in a medium saucepan, Over medium heat, melt the butter and bring to a simmer.
Remove the saucepan from the heat and vigorously stir in all of the flour with a wooden spoon until completely mixed in - taking care to mix in any flour stuck in the corners. It will resemble mashed potatoes at this point. Return the saucepan to the heat and cook the flour, stirring continually, for about 3 to 5 minutes. When done, the dough will pull from the sides of the pan and form a ball in the center. A film should have developed on the bottom of the pan, as well.
Transfer the dough to the bowl of an electric mixer. With the paddle attachment, cool the dough by mixing it on low to until it is just warm to the touch. Meanwhile, whisk 4 of the eggs together until smooth.
With the mixer on medium-low, begin adding in the eggs, a little at a time. You may only need 4 eggs, so be careful not to go too quickly. Continue mixing and adding in the eggs until the batter forms a "V" when pulled up out of the bowl with a spatula. Similarly, if you lift the paddle attachment, a tongue of batter should form on the end. The dough (or batter) should soft but not runny. It should be pipe-able, but still be able to hold it shape. Not sure if you added enough eggs? Try piping a small amount first. If necessary, whisk up the 5th egg and add a small portion of it at a time.
Fill a piping bag fitted with a plain round tip with the batter and pipe large mounds of batter - about 2-inches in diameter. Use a slightly dampened pastry brush or clean finger to gently press down any peaks that may form.
Remove the craquelin from the refrigerator. Use a biscuit cutter to cut rounds of the craquelin that are about the same diameter (or slightly larger) as the piped puffs. Top each puff with a craquelin “hat.” Store leftover craquelin in the freezer.
Bake the choux buns for 15 minutes at 400°F. Turn the oven down, and continue to bake at 350°F until golden (about 20 additional minutes). The puffs are done baking when the insides are dry and sound hollow when tapped from the bottom. Immediately use a wooden skewer or toothpick to poke the puffs and allow any trapped steam to escape. Cool before filling.