Dry the chicken breasts with paper towels and season both sides well with salt and pepper.
Combine the minced garlic, chopped parsley, and thyme in a small bowl, then rub the mixture evenly over each chicken breast, pressing it lightly so it sticks.
Let the seasoned chicken rest at room temperature for 10 minutes to absorb the flavors.
Heat the olive oil in a large skillet over medium-high heat until hot but not smoking.
Place the chicken breasts in the skillet and cook without moving them for 5-6 minutes to form a golden crust.
Turn the chicken over and cook on the opposite side for an additional 5-6 minutes.
Add butter to the pan and spoon the melted butter along with pan juices over the chicken breasts as they finish cooking.
Once the chicken reaches an internal temperature of 165°F, move it to a plate and cover loosely with foil to keep warm.
In the skillet with the remaining drippings, stir in the brown sugar and balsamic vinegar over medium heat until the sugar melts and starts to caramelize, about 2 minutes.
Pour in the chicken broth and scrape the bottom of the pan to loosen any browned bits.
Let the sauce simmer gently for 5-7 minutes until it thickens and becomes slightly syrupy.
Season the sauce with salt and pepper to taste.
Meanwhile, peel and chop the potatoes into even pieces and place them in a large pot with salted cold water.
Bring the pot to a boil, then lower the heat and simmer the potatoes for 15-20 minutes until soft and fork-tender.
Drain the potatoes well and return them to the pot.
Add butter and heavy cream, then mash the potatoes until smooth and creamy.
Season the mashed potatoes with salt and pepper according to taste.
Serve by placing a heap of mashed potatoes on each plate and top with a chicken breast.
Drizzle the reduced pan sauce over the chicken and around the plate.
Garnish with extra fresh herbs if you like and serve immediately while hot.
