Finely dice the onion into 1/8-inch pieces.

Heat the sesame oil in a large nonstick frying pan over medium heat. Fry the eggs one at a time until they reach desired level of doneness. Transfer eggs to a plate. Cover with foil or a second plate to keep warm.

Add the vegetable oil to the same pan over medium heat, then add the onion and cook until translucent, about 4 minutes. Remove from heat and set aside.

In a medium saucepan, combine the chicken broth, mirin, soy sauce, oil, and salt. Bring to a boil, then stir in the quinoa and reduce heat to a low simmer. Cover the pan and cook until liquid is absorbed and quinoa turns translucent, about 20 minutes. Remove from heat and stir in the sauteed onion. Cover and set aside.

Slice the mushrooms. Melt the butter in the frying pan over medium-high heat. Add the sliced mushrooms, salt, and pepper. Cook, stirring occasionally, until mushrooms begin to brown, 2-3 minutes. Transfer to a medium bowl and set aside.

Add the garlic oil and edamame to the frying pan and cook over medium heat, stirring occasionally, until heated through, about 1 minute. Add the salt and mix well.

Cut the seaweed into fine ribbons, and finely chop the green onion. To assemble, fill serving bowls with quinoa. Top with mushrooms, fried eggs, and edamame. Sprinkle roasted seaweed, green onion, and sesame seeds on top of the eggs. Drizzle each bowl with a few drops of toasted sesame oil and serve immediately.