Prepare the chicken and potatoes. Slice chicken breasts horizontally into thinner cutlets if they are thick. Pat completely dry with paper towels and season both sides generously with salt, pepper, and Italian seasoning. Wash and cube the potatoes into uniform 1-inch pieces.
Sear the chicken. Heat olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Once sizzling, add the chicken in a single layer (work in batches if needed) and sear for 4-5 minutes per side, until golden brown and cooked through. Remove the chicken from the skillet and set aside on a plate.
Sauté the potatoes. Add the remaining 1 tablespoon of butter to the same skillet. Add the cubed potatoes and sauté for 5-7 minutes, stirring occasionally, until they begin to brown and form a light crust.
Build the sauce base. Push the potatoes to one side of the skillet. Add the minced garlic to the clear area and cook for 30-60 seconds until fragrant, being careful not to burn it. Pour in the chicken broth to deglaze, scraping up any browned bits from the bottom of the pan with a wooden spoon.
Simmer the potatoes. Bring the broth to a simmer, then reduce the heat to medium-low. Stir in the heavy cream. Allow the potatoes to cook in the sauce for 10-15 minutes, simmering gently, until they are fork-tender and the sauce has thickened.
Add cheese and finish. Turn the heat to low. Slowly stir in the grated Parmesan cheese until it has completely melted. Next, stir in the shredded mozzarella. Return the seared chicken to the skillet, nestling it into the sauce. Spoon some sauce over the chicken and let it warm through for 1-2 minutes until the cheese is fully melted and bubbly. Garnish with fresh parsley and serve immediately.