Finely chop the cabbage and grate the carrots using the large holes of a box grater.
Heat olive oil in a large pan or cast-iron skillet over medium heat, then cook the chopped cabbage and grated carrots until they soften, about 5 to 6 minutes.
In a big bowl, whisk together the eggs, sea salt, black pepper, red pepper flakes, and soy sauce until combined.
Add the chopped parsley, sliced green onions, and grated mozzarella cheese into the egg mixture and stir gently.
Pour this egg mixture evenly over the cooked vegetables in the pan and cover with a lid.
Allow the omelette to cook over medium-low heat until the edges start to lift and only a few small wet spots appear on the surface, about 5 to 6 minutes.
Carefully slide the omelette onto a plate, flip the plate and pan together quickly, then slide the omelette back into the pan to cook the other side until golden, around 3 to 4 minutes.
Serve the omelette immediately, topped with extra chopped parsley and roasted sesame seeds if desired.
