Slice the chicken breasts in half horizontally to create four thinner cutlets. Place between two pieces of plastic wrap and pound to an even 1/4-inch thickness. Season both sides generously with salt and pepper.

Place the flour in a shallow dish. Dredge each chicken cutlet in the flour, ensuring a light, even coating. Shake off any excess.

In a large skillet, heat 1 tbsp olive oil and 2 tbsp butter over medium-high heat. Add two chicken cutlets and cook for 3-4 minutes per side, until golden brown and cooked through. Remove and set aside on a plate. Repeat with the remaining chicken.

Reduce the heat to medium and add the remaining 2 tbsp of butter to the skillet. Add the sliced mushrooms and cook until they have browned and released their liquid, about 5-7 minutes. Stir in the minced shallot and garlic, cooking for one more minute until fragrant.

Pour the dry Marsala wine into the pan to deglaze, scraping up any browned bits from the bottom. Let it simmer and reduce by about half, which should take about 2-3 minutes.

Stir in the chicken broth. Bring the sauce to a simmer and let it cook for 5-7 minutes, or until it has thickened slightly.

Return the chicken to the skillet. Spoon the sauce and mushrooms over the chicken and let it simmer for 1-2 minutes to warm through. Garnish with fresh parsley and serve immediately.