In a large bowl, toss the chicken with 1 tsp. each of the soy sauce and sesame oil. In another large bowl, beat the eggs with 1/4 tsp. salt, 1/2 tsp. pepper, and 1 tsp. of the sesame oil. In a small bowl, whisk the oyster sauce and vinegar with the remaining 1 Tbs. soy sauce and the remaining 1 tsp. sesame oil.
Heat 1 Tbs. of the canola oil in a large (12-inch) nonstick skillet over medium-high heat. Add the chicken and cook, stirring, until it loses its raw color, 2 to 3 minutes; transfer to a plate. Add 1 Tbs. of canola oil, the bell pepper and scallion whites and cook, stirring, until the vegetables begin to brown, about 2 minutes. Add the shiitake, sprinkle with 1/2 tsp. salt, and cook until softened, about 2 minutes. Add the vegetables and chicken to the egg mixture and stir gently to combine.
Wipe out the skillet with paper towels. Heat the remaining 2 Tbs. oil over medium-high heat until shimmering hot. Add the egg mixture, reduce the heat to medium, and cook, running a spatula in short, circular motions along the bottom of the pan, until the eggs begin to set, 2 to 3 minutes. Reduce the heat to low and gently press on the eggs with the spatula to flatten them. Cook until the eggs are just firm to the touch, about 2 minutes. Set a large plate over the pan and flip the eggs onto the plate so the bottom faces up. Drizzle with the oyster sauce mixture, sprinkle with the scallion greens, and serve.