Cover a cutting board completely with a piece of plastic wrap. Add chicken breast cutlets and cover with a second piece of plastic wrap. Pound cutlets to 1/4-inch thickness. Remove plastic wrap and season to taste with salt and pepper.
In a shallow dish, add 1/3 cup flour. Working with one cutlet at a time, dredge in flour and shake off excess. Transfer to a large plate and repeat with remaining cutlets.
In a large skillet over medium-high heat, add 2 tablespoons olive oil and 1 tablespoon butter. When the foaming subsides, add chicken in a single layer (you may have to work in batches). Fry until the chicken is golden brown on both sides and reaches 160 degrees on an internal thermometer, about 4 to 5 minutes per side. Transfer to a plate, tent with foil, and repeat with remaining chicken.
To the empty skillet, add 2 tablespoons olive oil and 1 tablespoon butter. Add the mushrooms and salt to taste (I like 1/2 teaspoon). Cook until lightly browned, stirring occasionally, about 6 to 8 minutes. Transfer to the plate with the chicken.
To the empty skillet, add remaining 1 tablespoon olive oil, shallots, and garlic and cook until softened, about 1 to 2 minutes. Stir in remaining 1 tablespoon flour and cook until thickened, about 2 minutes.
Stir in Marsala wine and chicken broth and cook, scraping up the browned bits from the bottom of the pan, until the sauce has thickened, about 2 minutes. Stir in the remaining 3 tablespoons butter until melted and season to taste with salt and pepper (I like 1/2 teaspoon salt and 1/4 teaspoon pepper).
Return the chicken and mushrooms to the pan with the sauce and toss to coat. Serve on a large platter family-style or plate smaller portions individually.