Prepare the Potatoes: Wash and peel the Yukon Gold potatoes. Slice them into consistent, 1/8-inch rounds (a mandoline slicer is ideal). Store sliced potatoes in cold water to prevent browning. Drain well before use.

Make the Sauce: Melt butter in a large saucepan over medium heat. Add onions and simmer until tender and transparent, 5-7 minutes. Add garlic and sauté until fragrant, about 1 minute.

Make a Roux: Whisk in flour and cook for 1-2 minutes. Gradually whisk in chicken stock until smooth.

Add Dairy: Gradually whisk in milk (or half-and-half) until smooth. Bring to a simmer, reduce heat to low, and cook, stirring occasionally, until the sauce has thickened slightly, about 5-7 minutes. It should coat the back of a spoon.

Season the Sauce: Remove from heat and stir in salt, pepper, and 1 teaspoon of fresh thyme leaves.

Layer the Potatoes: Preheat oven to 375°F (190°C). Grease a 9×13 inch baking dish. Layer potatoes, overlapping slightly, to cover the bottom. Pour 1/3 of the sauce over potatoes, then sprinkle with 1/3 of the cheddar cheese and 1/6 of the Parmesan cheese. Repeat layers two more times, ending with potatoes.

Top and Bake: Pour remaining sauce over potatoes, ensuring they are mostly covered. Sprinkle with remaining cheddar and Parmesan cheese. Cover with aluminum foil and bake for 30 minutes. Remove foil and bake for another 30-40 minutes, or until potatoes are tender and the top is golden brown and bubbly. Tent with foil if browning too quickly.

Rest and Serve: Let rest for 10-15 minutes before serving. Sprinkle with remaining fresh thyme and extra Parmesan cheese, if desired. Serve hot.