Combine flour, 3/4 tsp salt, and 1/4 tsp pepper in a zip-top bag. Add chicken, seal, and shake until evenly coated.

Heat olive oil and 2 tbsp butter in a large skillet over medium-high heat until butter stops foaming. Shake off excess flour from chicken and cook for 5 to 6 minutes, flipping once, until golden brown and just cooked through. Transfer to a plate and cover loosely with foil.

Add remaining 1 tbsp butter to the same skillet without wiping it clean. Add mushrooms and cook for 3 to 4 minutes, stirring frequently, until golden and moisture has evaporated.

Stir in shallots, garlic, and a pinch of salt. Cook for 1 to 2 minutes until shallots are translucent and garlic is fragrant.

Pour in chicken broth, Marsala wine, heavy cream, thyme, 1/4 tsp salt, and 1/8 tsp pepper. Scrape up browned bits from the pan. Bring to a gentle boil then reduce to a steady simmer. Cook uncovered for 10 to 15 minutes until sauce reduces by half and coats the back of a spoon.

Return chicken and any resting juices to the skillet. Reduce heat to low and simmer for 2 to 3 minutes until sauce clings to the chicken and everything is heated through.

Garnish with fresh parsley if using and serve immediately.