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; season the underside lightly as well.

Heat 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. Use tongs to lift and check the color.

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. Leave the rendered fat and flavorful 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.

Deglaze the pan: 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 browned fond into the liquid. Simmer for 1 minute until the liquid is reduced slightly.

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

Return Chicken to Skillet: Turn off the stovetop heat. Carefully place the seared chicken thighs back into the skillet, skin-side up, nestling them into the Dijon sauce. Do not submerge the crispy skin.

Bake: 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), checked in the thickest part of the thigh (avoiding the bone).

Rest and Thicken Sauce: 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.

Finalize Sauce: 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 parsley, and serve hot.