Heat the olive oil in a large skillet over medium-high heat.
Season the chicken cutlets on both sides with kosher salt and freshly ground black pepper.
Place the chicken cutlets in the hot skillet.
Cook the chicken until golden brown on both sides.
Transfer the browned chicken to a plate.
Cover the chicken to keep it warm and set aside.
Add the butter to the same skillet over medium heat.
Add the sliced yellow onions to the skillet.
Stir the onions until evenly coated with the melted butter.
Sprinkle the onions lightly with kosher salt.
Spread the onions evenly across the surface of the skillet.
Cook the onions for 30 to 35 minutes, stirring occasionally, until they are soft, sweet, and caramelized.
Reduce the heat as needed if the onions begin to brown too quickly.
Reduce the heat to low.
Add the Italian seasoning and minced garlic to the caramelized onions.
Cook for 30 to 60 seconds, stirring constantly, until fragrant.
Increase the heat to medium.
Add the dry white wine or dry rosé wine to the skillet.
Stir well, scraping the bottom of the skillet to loosen any browned bits.
Cook until the wine is mostly reduced.
Sprinkle the flour evenly over the onion mixture.
Cook for 2 minutes, stirring constantly.
Gradually stir in the low-sodium beef broth.
Simmer for approximately 5 minutes, or until the onion gravy is slightly thickened.
Return the browned chicken cutlets to the skillet.
Spoon some of the onion gravy over the chicken.
Top each chicken cutlet with an equal amount of grated Parmesan cheese.
Place 1 slice of Swiss cheese over each chicken cutlet.
Cover the skillet with aluminum foil.
Simmer gently until the chicken is fully cooked and the cheese is melted.
Remove the skillet from the heat.
Garnish with chopped fresh parsley, if desired.
Serve warm over noodles, rice, mashed potatoes, or preferred side dishes.