Slice the chicken breasts in half horizontally or pound them to an even thickness. Season both sides with salt and black pepper.
Place the flour in one shallow bowl, the beaten eggs in a second bowl, and combine the breadcrumbs with the grated Parmesan cheese in a third bowl.
Dredge each piece of chicken in the flour, dip it into the eggs, then coat thoroughly with the Parmesan breadcrumb mixture, pressing gently so the coating adheres well.
Heat the olive oil and 2 tablespoons of butter in a large skillet over medium heat. Cook the chicken for 4–5 minutes per side, or until golden brown, crispy, and cooked through to an internal temperature of 165°F (74°C). Transfer to a plate and keep warm.
In the same skillet, melt the remaining butter. Add the minced garlic and cook for about 30 seconds until fragrant.
Pour in the chicken broth and scrape up any browned bits from the bottom of the pan. Stir in the heavy cream and Italian seasoning, then bring to a gentle simmer.
Add the grated Parmesan cheese and stir until completely melted and the sauce is smooth and creamy. Season with salt and black pepper to taste.
Return the crispy chicken to the skillet and spoon the garlic butter cream sauce over the top. Simmer for 1–2 minutes to warm everything through.
Garnish with freshly chopped parsley and serve immediately with mashed potatoes, pasta, rice, or your favorite vegetables.