In a medium-sized saucepan set over medium-high heat, combine the water, butter, sugar, salt, and vanilla. Bring to a boil. Add the flour and stir until incorporated.
1¼ cups water, ½ cup unsalted butter, 4½ teaspoons granulated sugar, ½ teaspoon kosher salt, ½ teaspoon pure vanilla extract, 1¼ cups bread flour
Continue cooking and stirring the mixture until it begins coating the bottom of the saucepan in places. Remove from the heat.
Carefully transfer the mixture into the bowl of a stand mixer fitted with a paddle attachment.
Mix on a medium-low speed for a few minutes, or until the mixture has cooled slightly.
While the mixer is running, add the eggs one at a time, letting each one incorporate into the mixture before adding the next.
4 large eggs, 2 egg whites
Once all of the eggs have been added, continue mixing until smooth and glossy and the dough starts to hold a little shape.
In a Dutch oven, preheat 3-4 inches of oil to 375°F. Cut a piece of parchment paper into 3-4-inch squares.
Spoon the dough into a piping bag fitted with a very large open star tip. Pipe dough circles about 3-4 inches in diameter onto the parchment squares.
Place the piped dough circles with the parchment into the preheated oil. Gently remove the parchment paper with tongs and fry for about 2-3 minutes on each side or until a caramel golden color.
Once cooked and golden, carefully remove the crullers from the oil and place on a wire cooling rack over paper towels to let any excess oil drip out. Repeat the frying process with the remaining dough.