Heat the oven to 350°F with the rack in the middle position and grease a 2-quart gratin dish. Combine the sliced potatoes, carrots, and parsnips in a large bowl.

Melt the butter in a small saucepan over medium heat. Add the shallots and salt, cooking until softened, about 2 minutes.

Add the garlic, thyme, pepper, nutmeg, and cayenne. Cook until the spices are fragrant, about 30 seconds, then stir in the flour until it is fully incorporated into the butter.

Whisk in the cream and bring the mixture to a simmer. Cook until the sauce has thickened slightly, about 2 minutes.

Pour the warm cream sauce over the vegetables and toss until every slice is coated. Transfer the mixture to the prepared dish and press down gently to create an even layer and remove any air pockets.

Cover the dish tightly with aluminum foil and bake until the vegetables are almost tender, about 1 1/2 hours. A paring knife should meet some resistance when slipped into a potato slice.

Remove the foil and sprinkle the Gruyère evenly over the top. Return to the oven and bake uncovered until the top is lightly browned and a fork can easily pierce the center, 20 to 30 minutes.

Let the gratin cool for 10 minutes before serving. This allows the sauce to set so it doesn't run when scooped.