Season chicken with 1/4 teaspoon each pepper and salt. Place flour in a shallow dish and dredge the cutlets in it, pressing gently to fully coat.
Heat 1 tablespoon oil in a large skillet over medium-high heat until shimmering. Add half the cutlets and cook until lightly browned, 2 to 3 minutes per side. Transfer to a plate. Repeat with 1 tablespoon oil and the remaining cutlets. Cover to keep warm.
Add the remaining 1 tablespoon oil and creminis to the pan; cook, stirring occasionally until tender, 4 to 6 minutes. Transfer to a bowl. Add shiitakes to the pan and cook, stirring occasionally until tender, 4 to 6 minutes. Transfer to the bowl and season with the remaining 1/8 teaspoon salt.
Add onion, thyme and the remaining 1/4 teaspoon pepper to the pan; cook, stirring often, until the onion is translucent, 2 to 3 minutes. Return the mushrooms to the pan. Stir in Marsala and broth, scraping up any browned bits. Simmer over medium heat until slightly reduced, about 2 minutes. Remove from heat and stir in butter. Serve the chicken topped with the sauce.