Heat a large skillet or wok over medium-high heat and add 1 tablespoon of the olive oil. When hot, add the chicken pieces and cook, stirring occasionally, until browned and cooked through, about 5 to 6 minutes. Transfer the chicken to a plate and discard any accumulated juices from the skillet.
Reduce heat to medium and add 1 tablespoon olive oil. Add the sliced mushrooms, diced carrots, and diced bell pepper. Toss to coat and cook, stirring occasionally, until mushrooms are browned and vegetables are crisp-tender, about 8 minutes. Stir in the minced garlic and grated ginger and cook until fragrant, about 30 seconds. Transfer the vegetables to the plate with the chicken.
Add 1/2 tablespoon olive oil to the skillet. Pour in the beaten eggs and stir constantly with a spatula, breaking them into small curds. Cook until just set, then transfer the eggs to the plate with the chicken and vegetables.
Increase heat to high and add the remaining 1 1/2 tablespoons olive oil. Add the cold cooked quinoa and stir to coat with oil. Spread into an even layer and let it cook undisturbed until toasted and golden on the bottom, about 1 to 2 minutes. Using a spatula, flip large sections so uncooked portions contact the pan; repeat 1 to 2 more times until quinoa is evenly toasted.
Stir in the soy sauce, toasted sesame oil, and chili garlic sauce, mixing until quinoa is evenly coated. Add the cooked chicken, vegetables, eggs, any collected juices, and the thawed edamame. Stir to combine and cook over low heat, stirring occasionally, until warmed through, about 1 to 2 minutes.
Stir in the sliced green onions, taste, and adjust seasoning with more soy sauce or chili sauce if desired. Serve immediately.
