Add the sliced potatoes to a bowl and cover them with water. Wrap the bowl with plastic wrap and microwave for 5 minutes. Drain the water and set the potatoes aside. Preheat the oven to 410°F (200°C).
Melt the butter in a skillet over medium heat. Add the flour, then gradually stir in the milk. Continuously stir the mixture until the flour is fully dissolved. Season with salt, ground black pepper, and grated Parmesan cheese. Stir well to combine.
Add the potato slices to the white sauce and stir to coat them evenly. Turn off the heat. Transfer the coated potato slices to a baking dish and pour the remaining white sauce over them. Top with mozzarella cheese.
Bake for 10 to 15 minutes, or until the cheese melts and the top is golden brown.
Remove the potato gratin from the oven and let it rest for a few minutes before serving.