Pound and season the chicken. If using breasts, lightly pound to even thickness (about 1/2 to 3/4 inch).
Pat dry. Season both sides with salt, pepper, and paprika.
Dredge in flour. Place about half the flour on a plate. Lightly coat the chicken, shaking off excess.
Reserve the remaining flour for the sauce.
Sear the chicken. Heat olive oil and 1 tablespoon butter in a large skillet over medium-high. Add chicken in a single layer. Cook 4–5 minutes per side, until golden and just cooked through.
Transfer to a plate and tent with foil.
Cook the aromatics. Lower heat to medium. Add remaining butter. Stir in onions (and mushrooms if using) with a pinch of salt.
Cook 6–8 minutes, stirring, until soft and lightly browned. Add garlic and cook 30 seconds, until fragrant.
Build the roux. Sprinkle the remaining flour over the onions. Stir for 30–60 seconds to coat and cook off the raw flour taste.
Deglaze and simmer. Slowly whisk in the chicken broth, scraping up browned bits.
Add thyme, Dijon, and Worcestershire if using. Simmer 2–3 minutes until slightly thickened.
Add the cream. Reduce heat to medium-low. Stir in cream and bring to a gentle simmer.
Taste and adjust seasoning with salt and pepper. The sauce should be silky and spoon-coating.
Smother the chicken. Return chicken and any juices to the skillet, nestling into the sauce. Simmer 3–5 minutes, until heated through and sauce thickens to your liking.
Finish and serve. Squeeze a little lemon over the sauce if you like brightness.
Garnish with chopped parsley. Serve over mashed potatoes, rice, or egg noodles with a green vegetable or salad.