Place quinoa in a small saucepan, toast over medium heat (quinoa will turn golden brown and will pop as it is toasting.).
Stir in 3/4 cup water and 1/4 teaspoon salt. Bring to a boil, then reduce heat to low. Cover and cook until quinoa absorbs water, about 15 minutes. Remove from heat, fluff with a fork and leave uncovered.
Heat oil in a large skillet over medium-high heat. Cook carrot, stirring occasionally, until it softens, about 1 minute. Add bell pepper, ginger, garlic and chile, if desired; cook, stirring frequently, about 2 minutes. Add peas, sprinkle with remaining 1/4 teaspoon salt and pepper and cook, stirring frequently, 1 minute.
Remove vegetables and return skillet to heat; add quinoa, along with egg. Cook, stirring constantly, until egg is evenly distributed, about 2 minutes.
Add vegetables, chicken, scallions, cilantro and soy sauce; cook 1 minute more. Divide stir-fry among 4 bowls; serve warm.