Preheat your oven to 375°F (190°C) and grease a 9 by 13 inch baking dish.
Peel the 3 pounds potatoes and slice them about ⅛ inch thick. A mandoline makes quick work of this and keeps the slices even, which is what lets them all cook at the same rate.
Melt the 4 tablespoons butter in a saucepan over medium heat. Add the diced onion and cook for 3 to 4 minutes, until softened, then stir in the 3 cloves minced garlic for 1 minute more.
Sprinkle in the ¼ cup flour and whisk for 1 to 2 minutes, until it loses its raw smell. Slowly whisk in the 3 cups warm milk and cook, whisking, for 3 to 4 minutes, until thickened into a smooth sauce. Stir in the 1 teaspoon salt, ½ teaspoon black pepper, ½ teaspoon thyme, and ¼ teaspoon nutmeg.
Layer half the potato slices in the baking dish, overlapping them. Pour over half the sauce, then repeat with the remaining potatoes and sauce.
Cover tightly with foil and bake for 45 minutes. The foil traps steam so the potatoes cook through, which is the step that keeps them from staying firm.
Remove the foil, scatter the 1½ cups cheddar over the top, and bake uncovered for another 25 to 30 minutes, until the potatoes are fork tender and the top is golden and bubbly.
Let it rest for 10 minutes, then sprinkle with the 2 tablespoons parsley before serving. That rest lets the sauce set so it does not run everywhere.