Cook the quinoa according to the directions on the package. This will most likely call for 1 part quinoa to two parts water.
Heat 1 tablespoon vegetable oil in a skillet over medium heat for 1-2 minutes. Place the chicken in the skillet and cook until no pink remains, flipping at least once partway through the cooking time. Set aside to cool.
In a small saucepan, combine all the ingredients for the sauce, except the water. Heat over medium-low, stirring periodically, until smooth and slightly thickened, about 5-7 minutes. Be careful not to turn the heat too high, or the sauce will begin to sputter! After removing the sauce from the stove, stir in up to 3 tablespoons of water to thin it to the consistency of a salad dressing.
Once the chicken is cool enough to touch, chop or shred it into bite-sized pieces. Pour half of the peanut sauce over the chicken and stir until all surfaces are coated.
Cook the vegetables. You may do this however you prefer: steaming, stir-frying, parboiling, etc. I sautéed the bell pepper strips and snow peas with 1 tablespoon of vegetable oil for about 2-3 minutes, until just slightly softened, and steamed the broccoli in the microwave. To do this, put the broccoli in a microwave-safe bowl, place 1/3-1/2 cup water in the bowl, and microwave, covered, for about 2 minutes (the time will vary, depending on the power of your microwave).
Pour the other half of the peanut sauce over the quinoa and toss to combine. Add in the chicken and cooked vegetables. Garnish with peanuts, cilantro, green onions, and sesame seeds as desired.