Season chicken with salt and pepper. Mix balsamic vinegar, soy sauce, water, and brown sugar together in a small bowl until sugar is dissolved.

Heat oil in a large skillet over medium-high heat. Brown chicken in the hot oil until no longer pink in the center, 5 to 7 minutes per side. Remove from the skillet and set aside.

Add mushrooms and onion to the skillet, adding more oil as needed. Cook over medium heat until soft, 5 to 7 minutes. Return chicken to the skillet and pour balsamic glaze on top. Reduce heat to low and let simmer gently until sauce thickens slightly and chicken is warmed through, 5 to 7 minutes.