In a medium saucepan, combine 1 cup of water, ½ cup of unsalted butter, and 1 teaspoon of salt. Bring the mixture to a boil over medium heat, stirring occasionally.
Once boiling, remove the saucepan from heat and add 1 cup of all-purpose flour all at once. Stir vigorously with a wooden spoon until the mixture forms a ball and pulls away from the sides of the pan.
Let the dough cool for about 5 minutes. Then, add 4 eggs one at a time, mixing well after each addition until the dough is smooth and glossy.
Gently fold in 1 cup of shredded white cheddar, 1 cup of shredded medium cheddar, and ½ cup of minced scallions until evenly distributed throughout the dough.
Transfer the dough to a piping bag fitted with a large round tip, or use a zip-top bag with the corner snipped off. Pipe 1-inch rounds onto a baking sheet lined with parchment paper, spacing them about 2 inches apart.
Bake the gougeres in a preheated oven at 400°F (200°C) until golden brown.
