Lightly spray a large nonstick skillet with cooking spray and heat over medium.
Season both sides of the chicken breasts with salt and pepper.
Sear the chicken in the hot skillet until browned and cooked through (internal temperature 165°F), about 4–6 minutes per side depending on thickness. Transfer chicken to a plate and cover with foil to keep warm.
Reduce heat to medium-low and spray the pan again with cooking spray. Add the diced onion and sliced mushrooms, season with salt and pepper, and sauté until the onion is tender and mushrooms have released their moisture, about 5–7 minutes.
Pour in the Marsala wine and simmer until reduced by about half.
Add the chicken broth and stir to combine.
In a small bowl, mix the softened butter with the all-purpose flour to form a smooth paste (beurre manié). Whisk or stir the paste into the pan to thicken the sauce; simmer briefly until sauce is glossy and slightly thickened.
Return the chicken to the pan and heat through in the sauce for a minute or two, then serve hot.
