Preheat your oven to 375°F (190°C). Grease a 9x13-inch baking dish with olive oil.
In a large bowl, combine the heavy cream, minced garlic, thyme, salt, and pepper. Whisk until well mixed.
Layer half of the sliced potatoes in the bottom of the prepared baking dish. Pour half of the cream mixture over the potatoes, then sprinkle with half of the shredded cheese.
Place the chicken breasts on a cutting board. Season both sides with salt and pepper. Wrap each chicken breast with prosciutto slices, ensuring they are fully covered.
Place the prosciutto-wrapped chicken on top of the first layer of potatoes in the baking dish.
Layer the remaining potato slices over the chicken, then pour the rest of the cream mixture on top. Finish with the remaining cheese.
Cover the baking dish with aluminum foil and bake in the preheated oven for 30 minutes.
After 30 minutes, remove the foil and bake for an additional 20-25 minutes, or until the potatoes are tender and the cheese is golden brown and bubbly.
Let the dish cool for about 5 minutes before serving. Garnish with chopped parsley.