Pat chicken dry and season on both sides with salt and pepper.
In a large skillet, heat olive oil over medium-high heat. When the oil shimmers, add the chicken cutlets to the pan. Cook until the chicken is browned on both sides and the meat reaches an internal temperature of 165°F, about 3-4 minutes per side. Do not overcrowd the pan, so brown the chicken in batches if necessary.
Remove the chicken to a plate and set aside while you make the sauce.
Add butter to the skillet and melt over medium-high heat. When the butter melts, add the garlic, mushrooms, rosemary, and thyme. Cook, stirring occasionally, until the mushrooms are soft, about 6-8 minutes.
Add a pinch of salt to the mushroom mixture. Stir in the balsamic vinegar, chicken broth, and sherry.
Reduce the heat to low, and simmer uncovered for 12-15 minutes, allowing the sauce to reduce and thicken slightly. Taste the sauce and season with salt and pepper, if necessary.
Return the chicken cutlets to the skillet. Spoon sauce over top; cook on low just until the chicken is warmed through, about 1-2 more minutes.
Garnish with additional chopped fresh herbs, if desired. Serve immediately.