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To make the choux pastry: In a large saucepan, combine the water, butter, sugar, and salt. Place the saucepan over medium-high heat to melt the butter and bring the mixture to a boil.
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Remove the pan from the heat and add the flour all at once, stirring vigorously with a sturdy spoon or stiff spatula.
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Return the pan to medium heat and continue stirring until the mixture smooths out, begins to steam, and forms a crust-like coating on the bottom and edges of the pan, about 2 minutes. The batter will be very thick and shiny at this stage.
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Remove the pan from the heat and let the thick batter cool, stirring occasionally, for 5 to 10 minutes. It'll still feel hot, but you should be able to hold a finger in it for a few seconds. (The temperature should be below 125°F on a digital thermometer.)
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Finish the choux pastry, either by hand or with an electric/stand mixer.
To finish the choux pastry by hand: Add the eggs, egg white, and vanilla to the saucepan one at a time, incorporating each egg before adding the next. The mixture will slosh around a bit initially, but will come together into a smooth, shiny paste after a dozen or so stirs. Mix for about 2 minutes after adding the last egg.
To finish the choux pastry with an electric mixer: Alternatively, transfer the warm butter-water-flour mixture to a large bowl or the bowl of your stand mixer fitted with the flat beater attachment. Beat in the whole eggs, egg white, and vanilla one at a time on medium speed, waiting until each egg is fully incorporated before adding the next. Beat for about a minute after adding the last egg.
The choux pastry will be soft enough to pipe but firm enough to hold its shape.
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To pipe the crullers: On squares of parchment at least 4” in size, trace 3” circles, then turn the parchment upside down so the side with the traced marker is facing down. The circles will act as the template when piping your crullers.
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Prepare a pastry bag with a large star tip (see “tips,” below for our recommendations) and fill about halfway with choux pastry.
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Holding the pastry bag 2” or 3” above your traced circle, evenly pipe a circle of choux pastry. Be sure to overlap the ends of the circle. Repeat with the remaining choux pastry, refilling your piping bag as needed and piping each cruller on a clean parchment square with a traced circle on the back.
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Allow the piped crullers to rest, uncovered, while you heat the oil.
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To fry the crullers: In a deep pan, heat enough oil to fill the pot 3” deep to 375°F.
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Working with one parchment square at a time, pick up and gently place the cruller face-side down into the hot oil, parchment and all. Do not crowd your crullers; fry 2 or 3 at a time, depending on the size of the pan.
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After 20 to 30 seconds, the parchment circle will detach from the pastry. Once it does, remove the parchment with tongs and allow the cruller to fry for 4 to 6 minutes, until you see cracks forming on the surface. Discard the parchment.
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Gently turn the crullers over and fry for an additional 4 to 6 minutes.
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Use tongs or a slotted spoon to remove the crullers from the oil. Drain crullers on a wire cooling rack placed over a baking sheet, or onto a paper towel-lined baking sheet. Work in batches to fry the remaining crullers, being sure to maintain a consistent oil temperature of 375°F.
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To make the glaze: Choose 2 of the glaze flavors to finish all the crullers or make a double batch of one glaze if you want them to all be the same. In a small bowl, combine all the ingredients of your glaze(s) of choice, starting with the lesser amount of water. Stir and add more water as needed until you reach your desired consistency. A thicker glaze will fully coat the cruller and have an opaque finish, while a thinner glaze will easily coat the cruller and provide a shiny, thin coating.
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Dip the tops of the warm crullers in the glaze and serve as soon as possible. Crullers are best enjoyed fresh, the same day they are made.
