Preheat the oven to 400°F. Line two half-baking sheets with parchment paper.

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 the choux is ready. Add the mustard and mix for another 10 seconds to incorporate.

Sprinkle all of the cheese on top of the dough and mix by hand with silicone spatula. Scrape the sides and bottom the bowl to ensure all the dough and cheese are mixed together.

Use a 1-1/2-inch cookie scoop to scoop the choux dough onto the baking sheet about 2-inches apart. For a half-baking sheet there will be 15 on each sheet. Brush each gougere with the egg wash. Bake one baking sheet at a time.

Bake at 400°F for 10 minutes. Reduce heat to 350°F without opening the oven and bake for 15-17 minutes until choux buns are golden brown. 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.

The gougeres are best serve warmed. If making them for a party they can be made up to 2 days in advance and stored in the refrigerator. When ready to serve, preheat the oven to 350 degrees F and warm the gougeres for 5-7 minutes and until they are warm and crisp.

Freeze baked gougeres in a single layer on a sheet pan, then place them in a freezer bag. Reheat straight from the freezer in a 350 degree F oven until they are warm and crisp.