Season the chicken with 1 teaspoon of salt and ½ teaspoon of pepper.
Working one at a time, coat each chicken piece in the Parmesan mixture, ensuring an even, flavorful crust.
Heat butter and vegetable oil in a large cast iron skillet over medium heat.
Working in batches, add chicken to the skillet in a single layer. Cook until golden brown and fully cooked, reaching an internal temperature of 165°F, about 4-5 minutes per side. Remove from the skillet and set aside. Reduce heat to medium-low.
Stir in minced garlic and sauté until fragrant, about 1 minute.
Whisk in the remaining 1 ½ tablespoons of flour and cook until lightly browned, about 1 minute.
Gradually whisk in chicken stock, white wine, and fresh lemon juice. Bring the mixture to a boil, then reduce heat and let it simmer, stirring occasionally, until slightly thickened, about 3 minutes.
Stir in the remaining 2 tablespoons of Parmesan cheese, heavy cream, and chopped parsley. Season with salt and pepper to taste. Return the chicken to the skillet, coating it in the rich, creamy sauce.
Serve immediately, garnished with extra Parmesan and parsley if desired.