For the choux
Bring the milk, water, butter, salt and sugar to a boil over medium heat, stirring constantly. Add the flour all at once, stirring vigorously to combine. Cook the mixture over medium heat, stirring constantly until the mixture pulls away from the sides and a thin coating is formed on the bottom. Transfer the mixture to the mixer. Beat briefly on medium speed with the puddle attachment to cool the dough slightly. Add the eggs, a small amount at a time, and beat until fully incorporated. Add the emmental and graviera cheese and mix until fully incorporated. Using a pastry bag fitted with a plain tip, pipe the dough into 4 cm rounds onto a parchment lined sheet pan. Brush with egg wash. Sprinkle the parmesan on top. Bake at 190⁰C until golden brown, 10 to 12 minutes.