Pat chicken breasts dry with paper towels and season both sides generously with salt and pepper. Mix minced garlic, chopped parsley, and thyme in a small bowl. Rub the herb mixture all over each chicken breast and let rest at room temperature for 10 minutes.

Place peeled and cubed potatoes in a large pot, cover with cold salted water, and bring to a boil. Reduce to a simmer and cook for 15 to 20 minutes until completely fork-tender. Drain well, return to the pot, and steam dry for 1 minute. Add butter and warmed heavy cream and mash until smooth. Season with salt and pepper.

Heat olive oil in a large skillet over medium-high heat until shimmering. Sear chicken without moving for 5 to 6 minutes until a deep golden-brown crust forms. Flip and cook another 5 to 6 minutes. Add butter and baste the chicken with the pan juices. Once the internal temperature reaches 165 degrees F, transfer to a plate and tent with foil. Rest for 5 minutes.

In the same skillet over medium heat, add brown sugar and balsamic vinegar to the drippings. Stir constantly for about 2 minutes until the sugar dissolves and begins to caramelize. Pour in chicken broth and scrape up all browned bits from the pan. Simmer for 5 to 7 minutes until the sauce reduces by half and coats the back of a spoon. Adjust seasoning.

Spoon a generous portion of mashed potatoes onto each plate. Place one chicken breast on top and drizzle the pan sauce generously over the chicken and around the plate. Garnish with extra fresh parsley or thyme. Serve immediately.