Season the chicken well. Sprinkle both sides with salt, pepper, and paprika if using. Pat again to help the seasoning stick.
Blanch the green beans. Bring a pot of salted water to a boil.
Cook green beans for 2–3 minutes until bright green and just tender. Drain and rinse under cold water to stop the cooking. Set aside.
Sear the chicken. Heat 1 tablespoon olive oil in a large skillet over medium-high.
Add chicken and sear 4–6 minutes per side, until golden and nearly cooked through (165°F for breasts, thighs may take a minute longer). Transfer to a plate and tent with foil.
Sauté the mushrooms. In the same skillet, reduce heat to medium and add butter plus remaining olive oil. Add mushrooms in an even layer with a pinch of salt.
Cook undisturbed 2–3 minutes, then stir and cook another 4–5 minutes until browned and their liquid has mostly evaporated.
Add aromatics. Stir in onion and cook 2–3 minutes until softened. Add garlic and thyme; cook 30 seconds until fragrant.
Build the sauce. Sprinkle flour over the vegetables, stirring for 30 seconds to coat. Pour in chicken broth while stirring, scraping up browned bits.
Simmer 2 minutes to slightly thicken.
Finish it creamy. Stir in cream and Dijon. Bring to a gentle simmer. Taste and adjust with salt and pepper.
The sauce should be glossy and smooth.
Return chicken and add green beans. Nestle chicken back into the pan with any juices. Tuck green beans around it. Simmer on low 3–5 minutes, spooning sauce over the chicken, until everything is hot and the chicken is fully cooked through.
Garnish and serve. Sprinkle with parsley or chives.
Add a squeeze of lemon at the table for brightness. Serve with rice, mashed potatoes, or crusty bread.