Preheat the Oven:
Preheat your oven to 375°F (190°C). Grease a 9×13-inch baking dish with butter or cooking spray.

Prepare the Potato Layers:
Peel the potatoes and slice them thinly (about 1/8-inch thick) using a sharp knife or mandolin. Set aside.

Make the Creamy Sauce:
In a saucepan over medium heat, melt the butter. Add the minced garlic and thyme, and cook for about 1 minute until fragrant. Pour in the heavy cream and milk, stirring well to combine. Season with salt and black pepper, then bring the mixture to a gentle simmer. Let it cook for 2-3 minutes, then remove from the heat.

Layer the Potatoes:
Begin by layering the sliced potatoes in the bottom of the prepared baking dish. Overlap the slices slightly to create an even layer. Repeat the process until all the potatoes are used, layering them carefully.

Add the Cheese and Cream:
Pour the creamy sauce evenly over the potatoes, ensuring that it soaks into the layers. Sprinkle the grated Gruyere and Parmesan cheese evenly over the top. For an extra crispy topping, sprinkle breadcrumbs over the cheese (optional).

Bake the Gratin:
Cover the baking dish with aluminum foil and bake for 45 minutes. After 45 minutes, remove the foil and continue baking for an additional 15-20 minutes, or until the top is golden brown and crispy, and the potatoes are tender when pierced with a fork.

Serve:
Let the gratin rest for about 10 minutes before serving. Garnish with freshly chopped parsley if desired.