If you will be using a convection oven, move the oven rack to the middle position and heat the oven to 450°F.
Add 1 cup water, ¼ cup unsalted butter, 1 Tablespoon granulated sugar, and ¼ teaspoon salt to a medium saucepan over medium-high heat and bring to a boil.
Reduce the heat to medium-low and add 1 cup all-purpose flour. Stir the mixture with a rubber spatula, smoothing out any lumps of dry flour. Continue to cook and stir until the mixture is mostly smooth, with a thick, paste-like consistency, about 2-3 minutes.
Transfer the flour mixture to a mixing bowl and let it cool for 5-10 minutes. While you're waiting, cut 14-15 small (about 3-inch) squares of parchment paper. Set aside. Pour 2 Tablespoons neutral-flavored cooking oil or melted unsalted butter into a small bowl and get out a silicone brush (alternatively, use an oil sprayer if you have one).
Crack 2 large eggs into a small bowl or measuring cup and lightly beat them together with a fork.
Add the eggs to the bowl with the flour mixture. Beat together with a hand mixer until the egg is fully incorporated. Make sure to hold your mixer and the bowl firmly, as the batter will be very stiff (alternatively, you could use a stand mixer). Note that when you first add the eggs, the batter will intitially separate and look curdled. Keep mixing until the batter comes together. It will still be quite thick and pasty.
Scoop the batter in about 2 Tablespoon portions into the centers of the parchment squares. Brush each of the batter portions lightly but completely with the oil or melted butter. Place the parchment squares containing the batter into your air-fryer or onto a baking sheet at least 2 inches apart (depending on your air-fryer, you may need to cook them in two separate batches). While they're cooking, put 1/2 cup granulated sugar into a medium bowl.
Air-fry or bake for 10-14 minutes, until the edges are golden brown. Immediately brush all of the puffs lightly again with the oil or melted butter, then shake one or two pieces at a time gently in the bowl of sugar until the sugar is coating the outsides of the puffs. Repeat with all of the remaining puffs.
Serve immediately, as the puffs will turn soggy a few hours later, especially if you put them in an airtight container. If you need to reheat the puffs, put them in the air-fryer at 350°F for about 3-5 minutes.