Rinse quinoa thoroughly and combine with 2 cups of water in a medium saucepan; bring to a boil, then reduce heat to low and simmer covered for 15 minutes until liquid is absorbed.
Season chicken breasts with olive oil, salt, and black pepper.
Preheat a grill pan or skillet over medium-high heat and cook chicken for 6 to 7 minutes per side or until the internal temperature reaches 165°F (74°C).
Transfer chicken to a cutting board and allow it to rest for 5 minutes before slicing into 1/2-inch strips.
In a small mixing bowl, whisk together the peanut butter, soy sauce, rice vinegar, honey, grated ginger, and minced garlic until smooth; add 1-2 tablespoons of warm water if needed to reach desired consistency.
Divide the cooked quinoa evenly among four serving bowls.
Arrange the sliced chicken, shredded cabbage, julienned carrots, sliced bell pepper, and edamame on top of the quinoa base.
Drizzle each bowl with the peanut dressing and garnish with chopped peanuts and cilantro.
Serve immediately with lime wedges for squeezing.