Pierce the eggplant with a fork to let steam out as it cooks, preventing it from bursting.
Grill them on the stove over medium-high heat, preferably on a wire rack. Turn them with tongs every few minutes until the skin is charred and flaky, and the flesh becomes soft. Alternatively, you can broil them in the oven until charred.
When it is cool enough to handle, peel the charred skin while keeping the flesh intact. You can use a spoon to scrape any burnt bits, or give it a gentle rinse under running water.
Flatten them with a fork, keeping them as whole as possible. Then, set them aside.
Combine the eggs, fish sauce, a pinch of salt, some ground pepper, scallions, and chilies (if using) in a bowl. Beat until frothy to create a lighter and fluffier omelet.
Dip the eggplant into the egg mixture, coating them well. You can fry them together or one at a time to make it easier to handle in the pan.
Heat a small skillet or wok over medium-high heat and add a generous amount of oil, about ½-inch thick. Once the oil starts smoking, carefully slide the eggplant into the skillet. The hot oil will create a fluffy omelet with crispy edges. Fry until the edges are browned, about a minute. If the eggs have spread out too much, you can fold the sides inward.
Carefully flip the omelet and continue frying the other side for another minute until browned and thoroughly cooked.
Drain the excess oil as you lift it off the pan and place it on paper towels. Serve immediately and enjoy with ketchup, sweet chili sauce, or Sriracha.