Preheat the oven to 400°F. Line a large baking sheet with parchment paper and set aside.
Combine the butter, water, sugar, and salt in a medium saucepan. Place over high heat and bring to a rolling boil. Butter should have melted.
Take the pan off the heat and add the flour all at once. Stir vigorously with a wooden spoon for 2 minutes to form a dough. The dough will resemble mashed potatoes when ready.
Transfer the dough to the bowl of a stand mixer with a paddle attachment or a large bowl using a hand mixer. Beat on medium-low speed to let the heat out of the dough until the dough is just warm to the touch.
With the mixer on medium-low, add the eggs one at a time and fully mixing in each one before adding the next. Scrape down the sides of the bowl as needed between each addition.
Transfer the dough to a pastry piping bag fitted with a large round piping tip and pipe the dough in even 3-inch strips. Use clean kitchen shears to snip the dough near the tip when you're done piping each segment to reduce "tails". Make sure strips of dough are 2 inches apart.
In a small bowl, beat together the egg yolk and the teaspoon of water and brush the tops of the dough with egg wash: This makes the pastry golden.
Bake for 8 minutes at 400°F, then turn down the oven temp to 350°F and bake for another 20 to 25 minutes. Exact baking time will depend on the size and shape of your pastries. Bake until the pastries are slightly puffed, golden-brown in color, and dry to the touch. They will detach easily from the parchment and feel hollow and light when you pick them up.
Remove from the oven and "test" one of the pastries, if it still seems a bit soggy/eggy, lower the heat to 300°F and bake for another 5 minutes to help dry out the puffs
Poke an end of each pastry with a toothpick to release any lingering steam from the inside and help prevent them from getting soggy.
Cool the pastries completely on a cooling rack and bake off the remaining dough.