Crack the eggs into a mixing bowl. Add salt and milk (or water) and whisk until well combined and slightly frothy.

Prepare the vegetables by finely chopping the carrot, green onion, and bell pepper (if using).

Mix the chopped vegetables into the egg mixture, ensuring even distribution.

Heat the vegetable oil in a non-stick skillet over medium-low heat. Swirl to coat the surface evenly.

Pour a thin layer of the egg mixture into the skillet, enough to cover the bottom.

Cook until the bottom is set but not browned. Gently roll the omelette from one end to the other using a spatula or chopsticks, leaving a bit of space unrolled at the end.

Push the rolled egg to one side of the skillet and pour a bit more egg mixture into the empty side, letting it flow underneath the rolled omelette.

Once the new layer is about to set, roll the omelette further to incorporate it. Repeat this process until all the egg mixture is used, forming a thick roll.

Remove from the skillet when fully cooked, allow it to cool slightly, then slice into 1-inch thick pieces and serve warm.