Cut chicken into cutlets and pound to ¼-inch thickness.
Place 3 tablespoons flour, salt, and pepper in a zip-top bag. Add chicken, seal tightly, shake to coat evenly and set aside.
Heat olive oil and 2 tablespoons butter in large skillet over medium-high heat.
Place chicken in skillet, shaking off any excess flour. Cook, turning once, until golden and barely cooked through (about 5-6 minutes). Transfer to a plate and set aside.
Melt remaining 2 tablespoons butter in same skillet. Add mushrooms. Cook while stirring frequently, until brown (about 3-4 minutes).
Add 1 tablespoon flour and cook for 1-2 minutes.
Add broth, Marsala, and heavy cream. Scrape any brown bits from the pan into liquid. Bring to a boil. Reduce heat to medium and gently boil, uncovered, until sauce is reduced by about half, slightly thickened, and darkened in color (10-15 minutes). Sauce will be thin.
Turn off heat. Remove 1 cup sauce and add it to a small mixing bowl. Slowly whisk in French Onion Dip. Slowly stir back into sauce in skillet.
Add chicken with any accumulated juices. Turn heat to low, cover, and simmer until chicken is warm and sauce thickens (about 2-3 minutes). Garnish as desired and serve immediately.