Place one cutlet on a sheet of waxed paper. Cover with a second sheet of waxed paper, and pound cutlet to an even thickness with a mallet. Repeat with remaining cutlets.

Toss flour, 1/2 teaspoon of the salt and the pepper on a sheet of waxed paper. Add one cutlet, turning to coat. Repeat with remaining cutlets. Reserve 2 teaspoons excess flour mixture.

Heat oil and 1 tablespoon of the butter in a large skillet over medium-high heat just until sizzling. Add half of the cutlets and brown about 1 minute per side. Transfer cutlets to a plate; keep warm. Repeat with remaining cutlets, adding an additional tablespoon butter.

Reduce heat to medium and add remaining tablespoon butter. Stir in onions and cook 3 minutes. Add sliced mushrooms; cook another 3 minutes, stirring. Sprinkle with reserved 2 teaspoons flour mixture. Stir to coat. Add marsala and chicken broth. Bring to a simmer, stirring up browned bits from bottom of pan.

Add cutlets back to pan. Season with remaining 1/4 teaspoon salt. Cook, covered, 3 minutes, until sauce is slightly thickened. Serve over warm egg noodles.