Bring the bone broth, soy sauce, garlic and ginger to a boil in a medium saucepan. Add the oats and reduce the heat to medium low. Simmer, stirring occasionally, until the oats are softened and the liquid has reduced, 10 to 13 minutes.
Meanwhile, in a large, nonstick skillet, heat 1 tablespoon of the olive oil over medium until shimmering. Add the shiitake mushrooms and ½ teaspoon salt and a few grinds of pepper. Cook, stirring occasionally until golden brown, 8 to 10 minutes. Transfer the shiitakes to a plate and reserve.
Set the skillet back over medium heat and add the remaining 1 tablespoon olive oil. Add the eggs and cook until the edges begin to set, about 1 minute. Cover the skillet and cook until the whites are set but the yolks are still bright orange, 1 to 2 minutes.
Divide the oatmeal among 4 bowls. Top with the mushrooms, eggs, scallions, a sprinkling of sesame seeds and a drizzle of sesame oil. Top with your favorite hot sauce or chili crisp if desired.
