Preheat your oven to 425°F. Line a baking sheet with parchment paper or a silicone mat and set it aside.
In a big ol' saucepan over medium-high heat, combine butter, water, sugar, and salt. Bring it to a boil, then turn the heat down low and stir in the flour. Keep stirring with a wooden spoon until the dough pulls away from the pan. Cook it a bit longer (about 1 minute) to remove extra moisture.
Quickly transfer the dough to your stand mixer and mix on medium to let out some steam. Add vanilla, then eggs, one by one, mixing well after each addition. Mix until the dough is smooth and forms a thick ribbon when stirred.
Pop the dough into a piping bag with a ½" round or star tip. Pipe 1½-2" rounds onto your baking sheet, spacing 'em about 2" apart. Dampen your finger to smooth out the tip of each round for even baking and color.
Bake for 10 minutes at 425°F, then reduce the heat to 325°F (without opening the oven door) and bake for another 15-20 minutes until golden brown.
Take 'em out of the oven and let 'em cool on a wire rack.