Season the chicken: Pat the chicken thighs dry. Season all over with salt and pepper.
Dry skin crisps better, so don’t skip this step.
Brown the thighs: Heat the oil in a large skillet over medium-high heat. Place thighs skin-side down and cook undisturbed until the skin is deeply golden and crisp, about 7 to 8 minutes. Flip and sear the other side for 3 to 4 minutes.
Transfer to a plate; the chicken will finish cooking in the sauce.
Sauté mushrooms and aromatics: Pour off excess fat, leaving about 1 tablespoon in the pan. Add mushrooms and a pinch of salt. Cook, stirring occasionally, until they release moisture and get browned edges, 5 to 7 minutes.
Add onion (or shallots) and cook 2 to 3 minutes until softened. Stir in garlic, thyme, and oregano, and cook 30 seconds until fragrant.
Deglaze: Pour in the white wine (or 1/2 cup broth). Scrape up the crispy bits from the pan with a wooden spoon.
Let it simmer for 1 to 2 minutes to reduce slightly and cook off the alcohol.
Build the sauce: Stir in 1 cup broth, cream, and Dijon. If you want a slightly thicker sauce, whisk 1 teaspoon cornstarch with 1 tablespoon cold water, then add to the pan. Bring to a gentle simmer.
Return the chicken: Nestle the thighs back into the skillet, skin-side up.
Reduce heat to medium-low. Simmer gently until the chicken is cooked through (165°F at the thickest part) and the sauce has thickened, 12 to 15 minutes. If using Parmesan, stir it into the sauce in the last 2 minutes.
Finish and serve: Taste and adjust salt and pepper.
Add a small squeeze of lemon if you like brightness. Sprinkle with parsley. Serve hot over mashed potatoes, rice, or buttered noodles, with the creamy mushrooms spooned over the top.