Bring a large saucepan of salted water to a boil; add rice. Boil until rice is al dente, 30 to 35 minutes. Drain and rinse well with cold water; drain again.
In a large nonstick skillet, heat 1 teaspoon oil over medium. Add eggs; season with salt and pepper and cook until set, 1 to 3 minutes. Transfer to a cutting board and let cool; roll up and thinly slice crosswise.
In same skillet, heat remaining tablespoon oil over medium-high. Add mushrooms, garlic, ginger, and red-pepper flakes; season with salt. Cook, tossing frequently, until shiitakes are tender, 2 to 4 minutes. Add rice, eggs, edamame, scallions, lime juice, and soy sauce. Cook, tossing, until rice is heated through, 2 to 3 minutes.