Place each chicken breast between plastic wrap or in a zip-lock bag and pound to 1/4-inch thickness. Pat dry with paper towels.
In a shallow bowl, combine flour, salt, pepper, and oregano. Dredge each chicken breast in the flour mixture, shaking off excess.
Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Sear chicken 3-4 minutes per side until golden, then transfer to a plate.
Reduce heat to medium. Melt remaining butter, add mushrooms in a single layer, and cook 4-5 minutes until golden. Add shallot and cook 2 minutes, then add garlic and cook 30 seconds.
Pour in Marsala wine and scrape up browned bits from the pan. Simmer vigorously for 3-4 minutes until reduced by half.
Stir in chicken broth, heavy cream, brown sugar, Dijon mustard, and Worcestershire sauce. Bring to a gentle simmer and season with salt and pepper to taste.
Nestle the chicken back into the skillet, spoon sauce over the top, and cover. Simmer on medium-low for 10-15 minutes until chicken reaches 165°F and sauce has thickened.
Garnish with fresh parsley and freshly grated Parmesan. Serve immediately over pasta, mashed potatoes, or rice.