Preheat your oven to 375°F (190°C). Grease a 9x13-inch baking dish generously with 1 tablespoon of butter.

Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add the chicken pieces, season with 1/2 teaspoon of salt and 1/4 teaspoon of black pepper, and sear for 4-5 minutes until lightly browned but not fully cooked through. Remove from heat and set aside.

In a medium saucepan or microwave-safe bowl, combine the heavy cream, whole milk, minced garlic, fresh thyme, remaining 1/2 teaspoon of salt, and remaining 1/4 teaspoon of black pepper. Heat gently over low heat for 5 minutes until warm and fragrant, being careful not to let it boil.

Arrange half of the thinly sliced potatoes in an even layer at the bottom of the prepared baking dish, overlapping them slightly. Scatter the seared chicken pieces evenly over the potato layer.

Top the chicken with the remaining potato slices, creating a neat, layered top. Pour the warm cream mixture evenly over the entire dish, ensuring all the potatoes are moistened.

Sprinkle the grated Gruyère and Parmesan cheeses evenly over the top of the potatoes. Cover the baking dish tightly with aluminum foil to prevent the cheese from burning.

Bake in the preheated oven for 40 minutes. Then, carefully remove the foil and bake for an additional 15-20 minutes, or until the potatoes are tender when pierced with a knife and the top is deeply golden brown and bubbling.

Remove the gratin from the oven and let it rest for 10-15 minutes before slicing. This resting time allows the cream sauce to thicken and set, making it perfect to serve alongside some steamed green beans or a fresh garden salad.