Combine 1 cup water, 6 tablespoons unsalted butter, 2 teaspoons granulated sugar and 1/4 teaspoon kosher salt in a medium, heavy-bottomed sauce pan. Bring to a boil over medium-high heat, and add 1 cup all-purpose flour all at once, stirring until the flour is incorporated and the mixture is thick and pale yellow. Continue cooking over medium-high heat to get a much moisture to evaporate from the dough. A thin gluten coat will form along the bottom and sides of the pan. When enough moisture has evaporated, the dough will steam and the cooking flour will smell slightly nutty. To help facilitate this process, try and methodically smear the mixture across the bottom and sides of the pan.
Immediately transfer the mixture to the bowl of an electric stand mixer fitted with a paddle attachment. Beat on medium speed for about 1 minute, releasing some of the heat from the cooked dough. Add 3 large eggs, one at a time, only adding the next egg when the first one is fully incorporated. Stop the mixture between each egg addition and scrape down the bowl. Add 2 large egg whites, one at a time, until fully incorporated. The final mixture should be glossy and thick, holding its shape a little bit when picked up with a spoon but still just pourable.
Transfer to a piping bag with a 1 1/2-inch star tip and set aside. Prepare your deep frying vessel (see baker's notes) by pouring at least 2 inches of oil into your vessel of choice and attaching a deep-fry candy thermometer to its side. Preheat the oil to 370 (F) degrees.
While the oil is heating, take the parchment paper and cut at least 18 parchment paper squares around 4 x 4-inches large. Lightly brush each square on one side with vegetable oil, before piping a 3-inch circle of dough on each one of the squares.
Once the oil has preheated, carefully place one of the circles of dough, paper-side up, into the hot oil. After a minute, use tongs and a sharp knife tip to gently peel the parchment off the cruller. When the cruller turns golden (about 2 minutes), flip it over and let it fry for another 2 minutes before removing it to drain on a cooling rack or paper towels. Repeat until all dough circles have been fried — depending on your deep frying vessel, you can probably fry up to 2 or 3 crullers at a time. Just make sure that your oil maintains its temperature at 370 (F) degrees.