In a small bowl, combine the balsamic vinegar, soy sauce, brown sugar, minced garlic, and dried thyme. Stir well and set aside.
Pat the chicken thighs dry and season both sides with salt and pepper.
Heat a large skillet over medium heat. Add the olive oil and swirl to coat the pan.
Add the chicken thighs and cook until browned and cooked through, about 8 minutes on the first side and 5 minutes on the second side, depending on thickness. Transfer the chicken to a clean plate and cover loosely to keep warm.
Clean the mushrooms and cut them in half.
Add the mushrooms to the same skillet. Sauté for 5–7 minutes, stirring occasionally, until browned and slightly tender.
Reduce the heat to medium-low. Pour the prepared balsamic sauce into the skillet and scrape up the browned bits from the bottom of the pan.
Bring the sauce to a gentle simmer. Cook with the mushrooms for about 5 minutes, stirring occasionally, until the sauce has reduced by roughly half.
Add the butter and stir until completely melted and incorporated into the sauce.
Return the chicken to the skillet and turn each piece to coat with the sauce. Simmer for a few minutes until the chicken is heated through.
Serve the balsamic chicken and mushrooms hot, spooning the mushroom pan sauce over the chicken.