Season the chicken with the salt and black pepper and brown it in the olive oil over medium-high heat, 3 minutes per side.
Put the mushrooms, onion and garlic in a 6-quart slow cooker and lay the chicken over them.
Whisk the broth with 1/4 cup balsamic vinegar and the Worcestershire and pour it around the chicken.
Cover and cook on low until the breasts reach 165°F, 3 1/2 to 4 hours.
Transfer the chicken to a board and slice it across the grain.
Whisk the cornstarch with the cold water, stir it into the mushroom liquid and cook on high for 8 minutes.
Stir in the remaining balsamic vinegar and return the sliced chicken just long enough to coat it.