Pat chicken dry, season with salt and pepper. Brown in oil and 2 tbsp butter over medium-high heat, 5–7 minutes per side; transfer to a plate.
Add remaining butter to pan; sauté mushrooms until golden, about 5 minutes. Add shallot and garlic; cook 1 minute.
Pour in Marsala, scraping browned bits; simmer 2–3 minutes to reduce slightly.
Stir in broth, cream, and thyme. Return chicken to pan, skin-side up. Simmer gently, covered, 20–25 minutes until internal temp reaches 165°F.
Remove chicken, simmer sauce uncovered 3–4 minutes to thicken. Spoon sauce over chicken, garnish with parsley.