Turn on stove burner to low heat.
Coat tamagoyaki pan with butter. Holding the pan's handle, turn the fry pan back and forth and side to side to ensure all the corners get covered.
Optional: Use some folded up paper towel to coat the sides of the pan with butter and remove any excess melted butter.
With a fork or chopsticks, put a small amount of egg on the surface of the pan. If the egg sizzles, the pan is hot enough.
Pour enough egg mixture in pan to make a thin layer. Slightly tilt the pan and forth and side to side so the egg coats the entire bottom of the pan.
Once the first layer of egg is half cooked, use a small spoon add tuna on one-third of the layer of the egg. Make a line of tuna from from the long side to the other long side.
Evenly sprinkle shredded mozzarella cheese on top of the tuna.
With a spatula, roll up the first layer of omelette towards you to carefully wrap fillings inside. Keep rolling until you reach the end of the pan closest to you.
Once completely rolled up, nudge the roll to the far side of the pan, away from you.
Recoat the exposed pan with butter (if needed).
Add the remaining egg mixture to the pan. Slightly lift up the roll with your spatula. Then, tilt the pan away from you so the egg mixture evenly coats the pan underneath the first omelette roll.
Then, set down the pan flat. Once the next layer of egg is half-cooked, repeat the same rolling process towards you from step #8.
Using your flat side of your spatula, gently press down on the top and sides to form the rolled omelette into a neat rectangular prism shape.