In a large mixing bowl, whisk together the eggs, milk, fish sauce, salt, and black pepper until well combined.
Gently fold in the cooked crab meat, green onions, and cilantro until evenly distributed.
Heat 1 tablespoon of vegetable oil in a large non-stick skillet over medium heat. Pour half of the egg mixture into the skillet, tilting the pan to spread it evenly.
Cook for about 3-4 minutes, or until the edges start to set. Carefully lift the edges with a spatula and allow any uncooked egg to flow underneath.
Once the bottom is set and lightly golden, fold the omelette in half and cook for another 1-2 minutes. Transfer to a plate and keep warm.
Repeat the process with the remaining egg mixture, adding the remaining tablespoon of oil to the skillet if necessary.
Serve the omelettes warm with lime wedges on the side.