Place the chicken breasts on a cutting board and cover with parchment paper or plastic wrap. Using a meat mallet or the bottom of a heavy skillet, pound the chicken until it is a uniform ½-inch thickness. Pat dry with paper towels.
Cut the chicken into 1- inch cubes and place in a large bowl with the 1 tablespoon of the avocado, the salt, pepper and tapioca flour. Toss until the chicken is evenly coated.
In a large nonstick skillet, heat the remaining 2 tablespoons of avocado oil over medium-high heat. When the oil is hot but not yet smoking, carefully add the chicken pieces in a single layer, working in batches taking care to not overcrowd the skillet, and cook until golden brown on both sides, about 3 minutes per side. Transfer the cooked pieces to a clean plate and repeat until all of the chicken is browned.
To the same skillet you cooked, add the snow peas, carrot, and onion. Cook, tossing occasionally, until the vegetables are slightly tender, about 3 minutes. Return the chicken and any of its juices back into the skillet and pour in the teriyaki sauce. Continue to cook, tossing occasionally, until the sauce has thickened and coats the chicken, about 3 more minutes.
Serve over prepared quinoa and top with green onions and toasted sesame seeds, if desired.