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Put the butter, salt, and water in a saucepan. Bring to a boil until the butter is melted on high heat. Keep stirring.
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Reduce the heat to medium and fold in the flour to form a dough. Keep cooking and stirring until the dough is dried out and coat the pan with a crust.
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Turn off the heat and scrape the dough in a large bowl. Allow to cool for 15 minutes.
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Break and whisk the eggs into the dough with a hand mixer until the dough is sanity and firm enough to shape.
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Scoop up 1 tablespoon of the dough and roll it into a ½-inch-diameter and 2-inch-long cylinder. Repeat with remaining dough to make 12 cylinders in total.
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Combine the cinnamon and sugar in a large bowl and dunk the cylinders into the cinnamon mix to coat.
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Arrange the cylinders on a plate and refrigerate for 20 minutes.
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Spritz the air fryer basket with cooking spray. Place the cylinders in the basket and spritz with cooking spray.
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Put the air fryer basket on the baking pan and slide into Rack Position 2, select Air Fry, set temperature to 375ºF (190ºC) and set time to 10 minutes.
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Flip the cylinders halfway through the cooking time.
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When cooked, the cylinders should be golden brown and fluffy.
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Serve immediately.