Pour 3 cups vegetable oil into a large wok or deep skillet; you want it to come up about 2 inches on the sides. Heat the oil over medium-high heat until it reaches 350°F. If you don't have a thermometer, you'll know it's ready when vigorous bubbles form around an inserted wooden chopstick.
Using a ladle, carefully and slowly add about 1/4 of the egg batter to the hot oil. The egg foo young should immediately bubble up and puff. Fry until the bottom is golden brown and crispy, about 2 minutes. Gently flip the omelet; using two tools like a spider and a slotted spoon can make this easier. Continue frying for another 2 minutes until the second side is also golden and crisp. If you see any pale spots, use your ladle to baste them with hot oil.
Remove the cooked omelet and place it on a paper towel-lined baking sheet to drain. Let it cool for 5 minutes. Repeat with the remaining batter, making 4 omelets in total. Add more oil to the pan between batches if needed to maintain the 2-inch depth.