This is a very easy preparation for what I find to be an addictive pleasure. A quickly made dough called pâte à choux — the dough used to make cream puffs — is loaded with diced apple, fried, and rolled in cinnamon sugar. The doughnuts make a wickedly good start to the day and also a surprisingly easy and impressive passed dessert after dinner.
In a small saucepan over high heat, combine the butter and water. When the butter has melted and the water comes to a boil, reduce the heat to medium and add the flour. Stir until the flour absorbs the water and becomes a paste. Continue cooking the flour for another 30 seconds or so. Remove from heat. Stirring rapidly, add the eggs one at a time, and stir until completely incorporated. Let the dough cool enough to handle.
Add the apples to the dough and stir until well combined. Invert a large plastic bag over your hand and scoop out the apple dough. Cut a ½-inch hole in the corner of the bag. In a bowl large enough to toss doughnuts, combine the sugar and cinnamon and stir to mix.
Line a bowl with paper towels, and heat the oil in a pan for deep-frying to 350 degrees. (The oil should only come up about 1/3 of the way up the sides.) Pipe the dough into the oil, cutting the dough off at roughly 2-inch intervals or as desired. (Or shape them using 2 spoons and drop them into the oil off the spoons.) Cook until golden brown and cooked through, about 3 minutes. Remove 1 doughnut, cut it open, and see if the center is set and warm. Remove the doughnuts from the oil to the paper towel-lined bowl to drain, then roll them in the cinnamon sugar. Serve immediately.