Crack the eggs into a small bowl. Add a pinch of salt and black pepper. Whisk vigorously with a fork until the yolks and whites are completely combined and slightly frothy.

Heat the olive oil in a non-stick skillet over medium heat. Add the chopped onion and mushrooms. Sauté for 3–4 minutes until the onions are soft and the mushrooms are golden brown. Sprinkle the oregano over the veggies and stir for 10 seconds to release the aroma.

Reduce the heat to low-medium. Pour the whisked eggs directly over the sautéed veggies in the pan. Swirl the pan slightly to evenly distribute the eggs.

Let the eggs cook undisturbed for about 1–2 minutes. When the edges start to firm up, gently lift them with a spatula and tilt the pan so the uncooked, runny egg runs underneath.

Once the eggs are mostly set but still slightly glossy on top, sprinkle the cheddar cheese over one half of the omelette. Carefully fold the other half over the cheese.

Cook for another 30 seconds to let the cheese melt inside. Slide the omelette onto a plate and serve hot.