Season the chicken. If using chicken breasts (versus chicken cutlets), preheat an oven to 375°F (190°C). For either: Season ½ cup flour with 1 tsp Kosher salt and ¼ tsp black pepper, then dredge the chicken on all sides. Be sure to shake off any excess flour.
Cook the chicken. Heat 2 Tbsp olive oil and 1 Tbsp butter in a large cast-iron skillet over high heat. When hot, add the chicken, spacing slightly apart. Cook cutlets for about 5 minutes per side, until deeply golden, then transfer to a plate and tent loosely with foil. For whole breasts, cook about 7 minutes per side, then transfer to a rimmed baking sheet and finish in a 375°F (190°C) oven for 5–10 minutes, depending on thickness. Set aside.
Brown the onions. Add an additional 2 Tbsp butter, then add thinly sliced onion or shallots and garlic. Cook until browned and fragrant, about 2-3 minutes.
Make the sauce. Deglaze the pan with ½ cup white wine and ½ cup chicken stock. Use a spatula to scrape any browned bits off the bottom of the pan, then add 1 cup heavy cream, ⅓ cup grated Parmesan cheese, and butter.
Add spinach. Add 5oz fresh baby spinach to the pan, season with ½ tsp Kosher salt, then stir until the cheese has melted and the spinach has wilted.
Return the chicken to the pan, tossing once to coat on each side, then garnish with chopped parsley or chives and an extra sprinkle of grated Parmesan. Serve immediately!