In a large bowl, whisk the eggs together until they're light and frothy. Season with a pinch of salt and black pepper.
Gently fold in the crab meat, bean sprouts, chopped onion, green onions, celery, and sliced mushrooms into the whisked eggs. Be careful not to overmix; you want everything evenly distributed.
Place a large non-stick skillet or frying pan over medium-high heat. Add about 2 tablespoons of vegetable oil, enough to generously coat the bottom of the pan.
Once the oil is hot (a drop of egg mixture should sizzle immediately), pour about 1/2 cup of the mixture into the pan for each patty. You should be able to cook 2-3 at a time, depending on the size of your pan.
Cook for about 3-4 minutes per side, or until the bottom is golden brown and the eggs are set. Carefully flip each patty with a spatula. The inside should still be slightly soft.
Once cooked, remove the patties and place them on a plate lined with paper towels to drain any excess oil.
In a small saucepan, whisk together the chicken broth, soy sauce, oyster sauce (if using), and cornstarch. Bring to a simmer over medium heat, stirring constantly, until the sauce thickens slightly. This should only take a few minutes.
Transfer the cooked egg foo yong onto plates, you can place it on a bed of white rice. Then drizzle the sauce.