Place the apple cider in a small saucepan over medium heat. Simmer until it has reduced to approximately 1/4 cup. This concentrates the flavor. Set it aside to cool slightly before making the glaze.
In a medium saucepan, combine the water, butter, sugar, and salt. Bring the mixture to a rolling boil over medium-high heat. As soon as the butter is melted and the liquid is boiling, add the flour all at once. Stir vigorously with a wooden spoon until a smooth dough forms and pulls away from the sides of the pan.
Transfer the dough to a bowl and let it cool for about 5 minutes. Add the eggs one at a time, beating thoroughly after each addition. The dough should be thick, smooth, and glossy. It is ready when it forms a “V” shape when lifted with a spatula.
Fit a piping bag with a large star tip. Cut several 3-inch squares of parchment paper. Pipe a ring of dough onto each parchment square. This makes it easier to drop them into the hot oil without losing their shape.
Heat 2 inches of vegetable oil in a heavy pot to 350°F (175°C). Carefully place 2 or 3 parchment squares dough-side down into the oil. After a few seconds, the paper will peel off; remove it with tongs. Fry for about 2-3 minutes per side until deep golden brown and puffed. Drain on a wire rack.
Whisk together the reduced apple cider, powdered sugar, cinnamon, and vanilla in a shallow bowl until smooth. While the crullers are still slightly warm, dip the tops into the glaze. Let them sit on a wire rack for the glaze to set before serving.