To prepare the egg foo young mixture: In a large bowl, whisk together the eggs and salt. Mix the cornstarch with the water to make a slurry, then whisk it into the eggs. Add the crab sticks, garlic chives, carrot, and mung bean sprouts, and mix until evenly combined.
To pan-fry the egg foo young: Heat a wok over medium heat and add about 1 tablespoon of the oil. Add a ladleful of the egg mixture and cook for about 2 minutes, until the bottom is set and golden brown. Carefully flip and cook for another 2 minutes, until golden and cooked through. Remove and repeat with the remaining mixture, adding oil as needed.
To make the gravy: In a pot over medium heat, add the chicken stock, soy sauce, oyster sauce, and sesame oil. Whisk and bring to a simmer. Mix the cornstarch with the water to make a slurry, then slowly stir it into the sauce. Cook until thickened.
To serve: Arrange the Egg Foo Young on a serving plate and spoon the warm gravy over the top. Serve immediately.