Heat a large sauté pan over medium-high heat. When it’s good and hot, add the olive oil, then add the sprouts cut side down. Allow to cook for several minutes until well browned, and then flip.
Add the chicken to the pan, and season with salt and pepper. Stir and cook until the chicken is well browned, and add the thyme. Continue cooking until the chicken is cooked through, and add the balsamic vinegar and walnuts. Continue cooking until vinegar is reduced and walnuts are toasted before serving.