Heat the water, butter, sugar, salt, and vanilla together in a medium saucepan over medium-high heat until boiling. The butter should melt into the liquid as bubbles break across the surface.

Work the flour into the boiling mixture with a sturdy spoon or whisk. Keep stirring until no dry flour remains.

Stir the paste over the heat until patches of starchy film appear on the pan bottom, then take the saucepan off the burner. The mixture should gather into a cohesive mass rather than remain fluid.

Move the hot paste carefully into a stand mixer bowl with a paddle attachment, or a large bowl for a handheld mixer, and beat at medium-low speed for about 3 to 4 minutes. The paste should release heat and become less steamy.

Beat in the whole eggs individually, followed by the egg whites individually, fully blending each addition before the next. The paste may briefly look broken before it comes together again.

Continue beating after the final white until the dough develops a smooth sheen. It should retain some shape instead of spreading like batter.

Fill a large stockpot with oil to a depth of 3 to 4 inches and heat it to 375°F, monitoring with a clip-on digital thermometer. Leave ample empty space above the oil for bubbling and expansion.

Cut parchment into 3 to 4-inch squares while the oil heats and arrange them on a baking sheet. Keep the squares flat so they support the piped rings.

Load the dough into a piping bag fitted with a very large open star tip. The tip should leave pronounced grooves in the dough.

Form a 3 to 4-inch dough ring directly on each parchment square. Join the ends neatly without building a thick mound at the seam.

Lower each parchment-supported ring gently into the oil with the dough facing down. The paper should remain accessible above the ring.

Use long tongs to dip the parchment into the oil and ease it away from the dough. Let the paper release rather than pulling against a sticky ring.

Cook each cruller for about 2 to 3 minutes on each side. Look for an expanded ring with a deep golden exterior.

Drain the cooked rings on paper towels over a wire rack and continue until all the dough is fried. Keep the finished crullers aside while preparing the glaze.

Whisk the confectioners' sugar, corn syrup, hot water, and vanilla in a medium bowl. The glaze should look uniform, without dry sugar pockets.

Immerse a warm cruller in the glaze and rotate it with a fork to make sure both sides receive a coating. The glaze should reach into the fluted edges.

Lift the ring with a fork or wide slotted spatula and let the excess coating fall back into the bowl, then repeat with the remaining crullers. A thin coating preserves the definition of the ridges.

Arrange the glazed rings on a wire rack and wait for the coating to set before serving. The surface should lose its wet appearance.