Preheat the oven to 365°F.

In a saucepan set to medium heat, combine the milk, cream, and garlic; season to taste with salt and pepper. Do not boil. Decrease the heat and simmer for another 5 minutes, then set aside. (Do not be shy on the salt—it will help flavor the potatoes and turnips.)

In a small pan over medium heat, melt 1 tablespoon of the butter and add the leeks. Cook for 7 to 8 minutes, until the leeks start to brown, stirring frequently; then set aside.

Spread the remaining butter around a 9-by-13-inch baking dish, covering the bottom and sides. Arrange half of the potatoes, then half of the turnips in the dish, lightly seasoning each layer with salt and pepper. Add 1 cup of the Gruyère cheese and the cooked leeks on top of this layer. Pour enough of the cream mixture over the top to just barely cover.

Repeat with the remaining potatoes and turnips, again seasoning lightly with salt and pepper. Top with the remaining cheese, and drizzle the rest of the cream mixture on top, making sure to moisten everything evenly.

Bake for 40 to 45 minutes, until the top is golden brown and the vegetables can be pierced easily with a sharp knife.