Heat 1 tablespoon vegetable oil in a skillet over high heat. Add the mushrooms and 1/2 teaspoon sugar and saute until golden, 3 minutes. Add the broth and 2 teaspoons soy sauce. Mix the cornstarch with 1/4 cup water; stir into the skillet and boil until thickened, 2 minutes.
Heat 1 tablespoon vegetable oil in another skillet over high heat. Add the onion and the remaining 1 teaspoon sugar and cook until tender, 4 minutes. Add the bacon and cook 2 minutes. Add the peas, half of the scallions and the remaining 1 1/2 teaspoons soy sauce; cook 1 more minute. Transfer to a bowl and combine with the beaten eggs.
Wipe out the skillet. Return to high heat and add 1/4 inch of vegetable oil. Add one-fourth of the egg mixture; spoon some of the hot oil on top and cook until puffy, 1 to 2 minutes. Flip and cook 1 minute, then drain on paper towels. Repeat to make 3 more omelets. Add the sesame oil to the mushroom gravy and pour over the omelets. Garnish with the remaining scallions.