Heat olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Add the chicken cutlets and sear until golden brown and cooked through, about 5 to 6 minutes per side. Remove from the skillet and set aside on a plate.

Reduce the heat to medium. Add the remaining 1 tablespoon of butter to the skillet along with the sliced onions and mushrooms. Sauté for 8 to 10 minutes until the onions are deeply caramelized and the mushrooms are tender. Add the minced garlic and cook for 1 additional minute.

Pour in the chicken broth to deglaze the skillet, scraping up any browned bits from the bottom. Let the liquid simmer for 3 minutes to reduce slightly.

Reduce the heat to low. Stir in the sour cream, salt, and black pepper until a smooth sauce forms. Do not let the mixture boil.

Return the seared chicken to the skillet, spooning the sauce over the top. Let it heat gently for 2 minutes, then garnish with fresh chopped parsley before serving.