Preheat oven to 425F, place 1 rack in the center and the other at the lower third of the oven.
In a medium saucepan, add milk, water, butter, spices, salt, nutmeg and dijon (if using). Bring to a boil over medium-high heat, ensuring butter is melted, stirring with the rubber spatula to prevent scorching.
Take pan off burner, and add all the flour at once. Quickly whisk the flour into the liquid, taking care to get all the lumps out - this will take about 30 seconds. If you have lumps at this stage, you cannot get rid of them, and will need to start-over.
Return the mixture to medium-high heat, and cook, stirring constantly with the rubber spatula to scrape the bottom, for about 1 to 2 minutes to cook the paste. It is done when it starts to stick and hiss.
Transfer paste to the bowl of the stand mixer fitted with the paddle, or to a large bowl and wooden spoon. On medium speed, beat the paste to cool it for about 30 seconds.
Turn down to low (or stop mixing if by hand) and add 1 egg. Return to medium speed and mix until incorporated. Repeat with 1 more egg. Stop mixer and scrape down paddle and sides.
Add 2 additional eggs as in the previous step, scraping down the bowl and attachment after the 4th egg. Mix to incorporate everything.
Now, test the choux: stop the mixer, and take the paddle off the mixer and dip into the choux. If it forms a "V" shape, it is ready. If it breaks short of a "V" shape, add in 1 tsp of milk or egg wash (see recipe below) at a time with the mixer on low. Re-test the "V" after each addition. It is important to not over-add liquid or egg now, since you cannot recover a choux that is too runny.
Add the cheese and mix briefly to incorporate.
Fill the pastry bag fitted with the 1/2" or 3/4" tip (or two soup spoons or cookie scoop). Pipe or dollop 1 1/2" rounds of the gougere batter on cookie sheets, giving them 1" of space and staggering the rows. This allows them to expand, and cook/brown evenly on all sides.
Brush each gougere with egg wash using the pastry brush. If any tails are poking up, from piping, gently press them down with the brush with egg wash. Sprinkle on the additional cheese.*
Bake for approximately 25 minutes, rotating the pans half-way through at the 12 to 13 minute mark. The gougeres are done when they are golden brown, puffed to about double their size, and are firm to the touch.
Cool slightly, then serve. The gougeres are great warm or cool. Re-heat in a 350F oven for a few minutes to re-warm if desired.**