Place the cutlets between two sheets of plastic wrap and pound to an even 0.5 inch thickness; pat dry with paper towels.
In a shallow dish, whisk the all-purpose flour, kosher salt, and black pepper together. Dredge each cutlet in the seasoned flour, shaking off excess.
Heat the olive oil in a large skillet over medium-high heat until shimmering. Sear the boneless skinless chicken breast cutlets until golden, 3–4 minutes per side, just cooked through (155–160°F in the center). Transfer to a warm plate and tent loosely with foil.
Reduce heat to medium and add the 1 tablespoon unsalted butter to the skillet. Add the cremini mushrooms and cook, stirring occasionally, until browned and their liquid evaporates, 6–8 minutes. Stir in the shallot and garlic and cook until fragrant, 1–2 minutes.
Pour in the dry Marsala wine, scraping up browned bits, and simmer until reduced and slightly syrupy, 2–3 minutes.
Add the low-sodium chicken stock and simmer until the sauce is reduced by about half and lightly coats the back of a spoon, 5–7 minutes.
Return the chicken and any accumulated juices to the pan, turning to coat, and simmer until warmed through and the sauce clings to the cutlets, 2–3 minutes. Off the heat, swirl in the 2 tablespoons unsalted butter until glossy. Sprinkle with fresh parsley and serve immediately.