Prepare the sauce: In a small bowl, whisk together the balsamic vinegar, honey, kosher salt, black pepper, and dried thyme. Set aside.
Season the chicken thighs with kosher salt and black pepper.
Heat the ghee in a large (12-inch) nonstick skillet over medium heat, swirling to coat. Add the chicken thighs in a single layer and cook, undisturbed, for 7 minutes. Flip the chicken pieces and cook until cooked through, about 7 more minutes. Remove to a plate and cover with foil to keep warm.
Add the mushrooms to the same skillet with a splash of water (about 2 tablespoons). Gently stir the mushrooms while also scraping the bottom of the skillet to remove the tasty brown bits that have stuck to it.
After the mushrooms have started reducing a bit in volume, stir in the garlic.
Cook the mushrooms and garlic, stirring often, until the mushrooms are tender and browned, and the liquid that they have released has evaporated, 5-7 minutes.
Add the sauce to the skillet. Simmer, uncovered, stirring occasionally, until the sauce thickens, about 5 minutes. Stir in the butter until it melts.
Add the chicken thighs back to the skillet, turning them to coat in the sauce. Cook for a couple more minutes to heat through.
Divide the chicken and mushrooms between four plates. Drizzle or brush with the pan juices and sprinkle with parsley. Serve immediately.
If using sugar-free honey, a serving has 349 calories and 7g carbs.