Prep the air fryer. Preheat the air fryer to 375°F for at least 10 minutes while you make the choux dough. Generously spray the air fryer's wire basket with cooking spray.
Simmer the butter, milk, water, sugar, and salt. In a medium saucepan, combine the butter, milk, water, sugar, and salt. Cook over medium-low heat, mixing continuously with a wooden spoon, until the butter melts completely and the mixture is combined. Once the butter has melted, increase the heat to medium-high and bring to a simmer.
Add the flour and cook until glossy and smooth. Remove the pan from the heat and mix in the flour all at once. Continue to mix until the mixture has a paste-like consistency, about 2 minutes, then cook over medium heat, mixing quickly and continuously, until the dough pulls away from the sides of the pan and the bottom of the pan is clean, 1 to 2 minutes. The dough should be glossy and smooth, but not dry.
Transfer the dough to a stand mixer and beat in the eggs. Use a rubber spatula to immediately scrape the dough into the bowl of a stand mixer fitted with the paddle attachment. Beat on low for 30 seconds to release some heat, then add the eggs, one at a time, adding the next egg only after the previous one has been fully incorporated, scraping down the bottom and sides of the bowl after each addition. Once the eggs are fully incorporated, increase the mixer to medium-high and beat for an additional 15 seconds.
Pipe the churros. Use the rubber spatula to scrape the dough into a piping bag fitted with an open star tip. Use the piping bag to pipe 4 ½ to 5-inch long churros, each between ¼- to ½-inch thick, directly on the air fryer basket, with 1 to 1 ½ inches of space in between each.
Air fry the churros. Air fry the churros for 10 to 12 minutes, or until golden brown and crispy. Place the air fryer basket on a wire rack for 5 minutes to cool slightly, then transfer the churros with your hands or a stiff metal spatula to the wire rack.
Make the cinnamon sugar coating and coat the churros. In a shallow bowl, whisk together the sugar and cinnamon for the coating. Pour the melted butter in a shallow bowl. Working quickly, dip a churro into the butter to coat, then place in the bowl with the sugar. Toss to coat and place on a serving plate. Repeat with the remaining churros.
Serve and store. Serve immediately. The churros are best immediately after assembling, but will keep in an airtight container at room temperature for up to 3 days. To re-serve, toast in the air fryer for 5 to 10 minutes until warm and crispy.