Place a rack in the center of the oven with a rimmed sheet pan on the rack below, then preheat to 375°F (190°C). Grease a 9x13-inch baking dish with 1 tablespoon of butter.
Thinly slice the onion, mince the garlic, chop the thyme and parsley, and grate the nutmeg.
Melt 3 tablespoons of butter in a saucepan over medium heat and cook the onion for 6 to 8 minutes until soft. Add the garlic and thyme and cook for 1 minute.
Stir in the flour for 2 minutes, then slowly whisk in the milk and cream and simmer for 5 to 6 minutes until thickened. Off the heat, stir in the salt, pepper, and nutmeg, then cover and keep warm on the lowest heat.
Peel the potatoes and slice them 1/8 inch (3 mm) thick, without rinsing, right before assembling.
Layer one-third of the potatoes in the dish, pour over one-third of the sauce, and repeat twice more, ending with sauce. Press the top layer down gently.
Cover tightly with foil, arched so it does not touch the potatoes, and bake on the center rack for 60 minutes.
Remove the foil and bake 30 more minutes until deep golden and a knife slides through the center easily. Broil for 2 minutes if the top is still pale.
Rest the dish for 10 minutes, sprinkle with parsley, and serve hot.