Preheat your oven to 400°F (200°C).
Line a baking sheet with parchment paper to prevent sticking.
In a mixing bowl, combine the cold water, vegetable oil, and flour. Mix until the flour is fully incorporated into a soft and elastic dough.
Add the beaten eggs, one at a time, mixing well after each addition until the dough is smooth and glossy.
Gently fold in the grated Romano cheese and freshly ground black pepper until evenly distributed throughout the dough.
Using a piping bag or spoon, drop tablespoon-sized amounts of dough onto the prepared baking sheet, leaving space between each for expansion.
Whisk together 1 egg and 1 tablespoon of water. Brush this mixture over the tops of the gougères for a golden finish.
Bake in the preheated oven for 20-25 minutes, or until puffed and golden brown. Avoid opening the oven door while baking.
Remove from oven and let cool slightly before serving. Enjoy warm for best flavor.