Coat chicken evenly in flour seasoned with salt and pepper. Set aside.
Heat olive oil and 1 Tbsp butter in a large skillet over medium high heat. Brown chicken, about 4-5 minutes each side. Remove from skillet and set aside.
Stirring frequently, cook mushrooms in remaining 1 Tbsp butter for about 4 minutes, or until browned.
Add shallots and garlic, cooking until shallots are fragrant and translucent, about 2 minutes.
Add chicken broth, Marsala, heavy cream, and thyme, stir and scrape up any bits from the pan.
Bring to a boil, and then reduce heat to medium and simmer, stirring occasionally, until sauce thickens and reduces by half. About 10-12 minutes.
Return chicken to the pan and heat through, about 2 minutes. Serve with cooked linguine and sprinkle with shredded Parmesan.