Preheat your oven to 400°F (200°C). Pat the chicken thighs thoroughly dry using paper towels. This step is critical for achieving crispy skin. Season the skin side of the chicken heavily with salt and black pepper, and season the underside lightly as well.

Place a heavy oven-safe skillet (cast iron or stainless steel) over medium-high heat. Add the olive oil and the butter. Allow the butter to melt and the fat to shimmer.

Carefully place the chicken thighs in the hot skillet, skin-side down. Reduce the heat slightly to medium. Let the chicken sear undisturbed for 7 to 9 minutes, or until the skin is deep golden brown and exceptionally crispy. Flip the chicken thighs and cook on the meat side for just 1 minute. Remove the chicken from the skillet and set it aside on a plate, leaving the rendered fat and browned bits (fond) in the pan.

Reduce the heat to low or medium-low. Add the minced garlic and fresh thyme to the skillet. Cook for about 30 to 60 seconds until fragrant, ensuring the garlic does not burn.

Pour in the dry white wine (if using) and/or a splash of chicken broth. Use a wooden spoon to scrape the bottom of the pan, dissolving the flavorful fond into the liquid. Simmer for 1 minute until the liquid is reduced slightly.

Pour in the remaining chicken broth and the Dijon mustard. Whisk continuously until the mustard is fully incorporated and the sauce is smooth and uniform. Season the sauce with the measured 1/2 teaspoon of salt and 1/4 teaspoon of black pepper.

Turn off the stovetop heat. Carefully place the seared chicken thighs back into the skillet, skin-side up, nestling them into the Dijon sauce. Transfer the skillet immediately to the preheated 400°F (200°C) oven.

Bake for 15 to 20 minutes, or until the internal temperature of the chicken thighs reaches 175°F (80°C) when checked with a meat thermometer, inserted into the thickest part of the thigh, avoiding the bone.

Remove the skillet from the oven. Transfer the cooked chicken to a cutting board or serving platter and tent loosely with foil to rest for 5 minutes.

Return the skillet with the sauce to the stovetop over low heat. If using, whisk in the heavy cream until the sauce is creamy and slightly thicker. Simmer gently for 1 to 2 minutes. Taste the sauce and adjust seasonings (salt/pepper) if necessary. Spoon the Dijon pan sauce generously over the rested chicken thighs. Garnish with fresh chopped parsley and serve hot.