For a lighter gratin that doesn't need to wait for winter, layer potatoes and carrots with parmesan and lots of fresh thyme and use chicken stock as your liquid. It's just as good at room temperature as it is hot, which makes it an unlikely but excellent picnic candidate.
Heat the oven to 375 F° and rub the inside of a 3-quart baking dish with 1 of the garlic cloves and 1 tablespoon of the butter.
Warm the stock with the second garlic clove and the sprig of thyme in a small saucepan over low heat.
Peel the carrots and potatoes and slice them very thinly (1/16th inch), using a mandolin or a very sharp knife; cut the carrots on the diagonal so that they’re as close in size to the potatoes as possible. If you're using a knife, put the sliced potatoes in a bowl of cold water to keep them from browning as you work; then dry them on a kitchen towel before you layer them in the dish.
Cover the bottom of the baking dish with a layer of alternating slices of potato and carrot; start at the edges and work your way towards the center of the dish, overlapping the slices. Season the vegetables with salt and pepper and sprinkle a third of the chopped thyme leaves and a third of the Parmesan evenly over them. Repeat with two more layers of potatoes, carrots, salt, pepper, thyme, and cheese. Take extra care with the top layer of vegetables—you want it to look nice, right?
Scoop the garlic and thyme out of the stock and carefully pour it over the vegetables. Pour the melted butter evenly over the top and bake for about an hour, until the gratin is brown and bubbly and a sharp knife slides easily through the layers. If you’d like more browning, slide the gratin under the broiler for a minute or two, keeping a close eye on it. Let the gratin sit at room temperature for 5 to 10 minutes before serving.