Heat a large skillet or wok over medium heat and add 1 tablespoon of olive oil. Add in the mushrooms and toss to coat, then cook, stirring occasionally until soft, about 5 to 6 minutes. Stir in the garlic and ginger and cook for another minute. Remove the mushrooms from the skillet and place them (and any juice) in a bowl. Increase the heat to medium and add 1/2 tablespoon olive oil. Add the beaten egg and cook, using a spatula the entire time to stir and break the egg into pieces. Remove it from the skillet and set it aside in a bowl.
Add the remaining olive oil to the skillet and increase the heat to high. Add in the rice and stir well to coat. Spread it into one layer and cook until it’s toasted and golden, about 1 to 2 minutes, then flip it and do the same one or two more times. Stir in the soy sauce and sesame oil. Add the mushrooms (and a little of the juice if desired) back into the rice along with the eggs. Stir well to coat. Stir in the sesame seeds and green onions, reduce the heat to low and cook until everything is warm and has come together. Taste and season additionally if needed – this will depend on your soy sauce. Serve immediately.