Preheat oven to 300 degrees F. Season chicken thighs all over with salt, pepper and nutmeg. Place skin-side down in a roasting pan and cover tightly with foil. Bake for 45 minutes. Remove foil, turn chicken pieces skin-side up and bake for 1 hour.
When ready to serve, broil chicken thighs for 10 to 15 minutes, until dark brown and crisp. Meanwhile, heat a cast-iron skillet over very high heat. Add carrots, cover and cook for 10 minutes, uncovering and stirring frequently, until charred and softened. Transfer to a serving bowl and toss with parsley and butter; season with salt, pepper and nutmeg. Heat demi-glace sauce to a simmer. Serve chicken and carrots, drizzled with sauce.