Preheat the broiler. Whisk the eggs in a bowl. Heat 1 tablespoon peanut oil in a medium nonstick ovenproof skillet over medium-high heat. Add the mushrooms and stir-fry 5 minutes. Stir in 1 teaspoon soy sauce. Transfer to the bowl with the eggs using a slotted spoon.
Add the snap peas, ginger and half of the scallions to the skillet; stir-fry until crisp-tender, 1 to 2 minutes. Add the shrimp and 2 teaspoons soy sauce; cook 1 minute. Add to the bowl with the eggs.
Whisk 1/4 cup water, the broth, cornstarch and the remaining 2 teaspoons soy sauce. Wipe out the skillet. Add 1 tablespoon peanut oil and the remaining scallions; stir-fry 30 seconds. Add the broth mixture, reduce the heat to medium low and simmer, whisking, until thick, 2 minutes. Stir in the sesame oil. Transfer to a bowl and cover.
Wipe out the skillet. Increase the heat to medium. Add the remaining 1 tablespoon peanut oil and the egg mixture; cook, undisturbed, 5 minutes. Broil until set, 5 minutes. Serve with the rice and sauce.