Season chicken with salt and pepper. Heat 2 tbsp oil in a large skillet over medium-high and brown chicken 3–4 minutes per side until cooked through; transfer to a plate.
Add remaining oil, sauté mushrooms until golden about 5 minutes, then add garlic and cook 30 seconds.
Pour in Marsala and simmer 2 minutes to reduce, then add chicken broth and thyme; stir and return chicken to pan to warm and finish sauce.
Meanwhile steam green beans in a covered skillet or steamer for 4–6 minutes until tender-crisp.
Swirl in butter to finish sauce and spoon over chicken; serve with green beans and lemon if liked.