Break the eggs into a bowl, add salt and mix well using an egg beater until the yolks and white are combined.
Add the ¼ cup of shredded cheese to this egg mixture, beat well again.
Heat the butter in a pan until the butter is foamy but no need to brown the butter.
Add the egg mixture to a small sized pan and give it few seconds to lightly set in the pan.
Keep the heat at medium to low, and stir the egg mixture continuously using a small non stick spatula or spoon for 1 to 2 minutes until the egg is set and not runny anymore. You can gently press the top of the omelette mixture using the spatula or spoon to make it even.
Add the remaining 1 cup of shredded cheese to the top of the egg mixture, cover the pan with a lid and cook in low heat for about 2 minutes or until the contents are cooked and the cheese is all melted. As the eggs is all set and cooked, you can see that the edges are no more sticking to the pan.
Gently transfer the cheesy pizza style egg omelette from the pan to a serving plate (using a large non stick spatula will help you to transfer the omelette easily).
Give the omelette 1 to 2 minutes to cool down, gently slice the omelette like you slice for pizza ( a good spatula can easily slice the omelette)and serve hot!
If you are serving the omelette for adults you may sprinkle the pizza style omelette with spices like crushed red pepper flakes, ground black pepper, oregano flakes etc.