Cut each chicken breast in half lengthwise. Pound each piece to about 1/4 inch thick between two sheets of plastic wrap using a meat mallet or rolling pin.

Heat 3 tablespoons olive oil in a large deep skillet over medium-high heat until shimmering. On a plate, mix together the flour, salt, and pepper. Dredge each piece of chicken through the flour mixture and shake off excess.

Cook chicken in batches without crowding the pan, 3 to 4 minutes per side, until deep golden brown and cooked through. Remove to a plate and set aside. Add more oil between batches if needed.

Reduce heat to medium. Melt butter in the same skillet. Add sliced mushrooms and diced onion. Cook for about 3 minutes, stirring occasionally, until softened and lightly browned.

Add the minced garlic and 1 tablespoon flour. Stir and cook for 1 minute. Scrape up any browned bits from the bottom of the pan.

Pour in the Marsala wine and simmer for 1 minute. Whisk in the chicken broth and heavy cream. Bring to a low boil, then reduce heat and simmer uncovered for 8 to 10 minutes, stirring occasionally, until the sauce thickens and turns a rich golden brown. If the sauce is still thin after 10 minutes, continue simmering for 2 to 3 more minutes.

Return the chicken to the skillet and heat in the sauce for 2 to 3 minutes until warmed through. Serve immediately over egg noodles, pasta, rice, or mashed potatoes.