Put the broccoli in a large microwave-safe dish; add 2 tablespoons water, cover with plastic wrap and microwave until bright green, about 2 minutes. Drain and set aside.
Heat 1 tablespoon vegetable oil in a large nonstick skillet over high heat. Add the mushrooms in a single layer and cook, undisturbed, until browned around the edges, about 4 minutes. Stir in the broccoli. Cook, stirring occasionally, until the broccoli is warmed through, about 2 minutes. Add the scallions, edamame, water chestnuts, bean sprouts, snow peas, teriyaki sauce and Sriracha. Continue to cook, stirring, until warmed through, about 4 minutes.
Divide the rice and vegetables among bowls. Wipe the skillet clean and heat the remaining 1 tablespoon vegetable oil over medium-high heat. Crack in the eggs and cook until the whites are set, about 2 minutes. Add a fried egg to each rice bowl; top with more Sriracha.