Preheat the oven to 400°F. Draw 2-inch circles, 2 inches apart on parchment paper and line two baking sheets with the pencil side down.
Sift the flour into a bowl or onto a piece of wax paper. Crack the eggs into a separate bowl.
Place the milk, water, butter, sugar and salt in a 4-quart saucepan. Bring the mixture to a boil over medium high heat and make sure all the butter is melted. Remove the pot from the heat and dump in all the flour all at one and quickly stir with a wooden spoon until there are no bits of flour left. You do not want any lumps of flour. Return the saucepan to medium heat and continue to cook the choux while stirring vigorously. The mixture will sizzle as it continues to cook, and a thin layer will form on the bottom of the pan. This step is to cook the protein in the flour and cook some of the moisture in the choux. The dough will become a smooth ball.
Put the dough into the bowl of a stand mixer and with the paddle attachment beat on medium speed for 1 minute to cool the choux pasty. Reduce the speed to low, add one egg and increase the speed to medium and beat until the egg is incorporated. Repeat with each egg. After add the second egg, stop the mixture and scrap the bottom and sides of the bowl. After the last egg is added beat the dough on medium speed for another 1-2 minutes until it is smooth. To test the dough stop the mixture, unhook the paddle attachment, push it into the dough and then slowly pull up. If the dough forms a V-shape at the bottom the attachment it is ready.
Add an 1/2-inch piping tube to an 18-inch pastry bag. Fill the pastry bag with the choux dough. Hold the bag straight up and down over the baking sheet about 3/4-inch from the surface and pipe a 2-inch round. There should be room for 12 on one baking sheet, and 6 on the other.
Bake one baking sheet at a time. Place one craquelin circle on top of each choux round right before it goes in the oven. Place the remaining craquelin back in the freezer.
If you're making the choux buns without the craquelin, wet the tip of your index finger and gently pat down the tip of the bun. Make an egg wash by whisking together one egg and 1 teaspoon of milk and gently brush the buns with the egg wash.
Bake at 400°F for 20 minutes. Reduce heat to 350°F and bake for 10-15 minutes until choux buns are golden brown and the craquelin has melted oven the choux bun. Remove them from the oven and place the pan on the cooling rack. Increase the oven temperature to 400°F and wait for it to reach full temperature before baking the second pan.