In a medium pot (use stainless steel and avoid nonstick pans), bring water, unsalted butter, and salt to a boil over high heat. Remove from heat and stir in flour with a wooden spoon, and then return to heat and cook while mixing until a skin forms on the bottom of the pot.
Remove from heat and transfer dough to a stand mixer fitted with a paddle attachment. Mix on medium speed until dough stops steaming and bowl is no longer hot to touch.
Add eggs one at a time, fully incorporating after each. The exact amount of egg you need will vary based on how much moisture was evaporated. Continue until dough falls off the paddle and leaves a "v" shape hanging from the paddle. Add the grated parmesan cheese and mix to incorporate.
Transfer dough to a piping bag fitted with a round tip (approx. 1/2in). Pipe dough into long lines on a sheet pan lined with parchment. Freeze until firm.
While dough is chilling, boil a pot of salted water and prepare sage brown butter.