Heat two teaspoons of the oil in a large skillet. Add onions and garlic and sauté, stirring often, until translucent, about 5 minutes. Add ground beef or pork and cook until cooked through, stirring often and breaking large clumps with a wooden spoon, about 8 to 10 minutes; season with salt and pepper. Drain the beef mixture and let cool for 10 minutes.
Add eggs and fish or soy sauce to the cooled beef mixture and stir to combine.
In an 8-inch non-stick skillet, heat one teaspoon of the oil over medium-low heat. With a ladle, scoop some of the mixture (about ¼) into the hot oil. Let omelet set for 3 minutes, then sprinkle with some of the green onions. Flip omelet and cook 2 minutes longer. Transfer to a plate. Repeat with remaining oil, egg-beef mixture, and green onions until the mixture is gone.
Serve omelet individually, or stack on top of each other and cut into 4 layered slices. Serve with ketchup or sriracha chili sauce.
To flip the omelet more efficiently, try upturning the skillet to flip it onto a plate and then transfer it back into the skillet with a spatula.