Preheat your oven to 375°F (190°C).
In a saucepan, whisk together the white miso paste, heavy cream, vegetable broth, minced garlic, thyme leaves, black pepper, a pinch of salt, and nutmeg. Heat the mixture over medium heat, stirring until the miso paste is fully dissolved and the ingredients are well combined. Remove the mixture from heat and stir in the onion jam..
In a large mixing bowl, combine the thinly sliced root vegetables with the miso cream mixture. Make sure the vegetables are well coated.
Grease 13 x 9 baking dish with olive oil. Layer the coated root vegetables in the dish, slightly overlapping the slices. Pour any remaining miso cream mixture over the top of the vegetables.
Cover the baking dish with aluminum foil and place on a baking sheet. Bake in the preheated oven for about 45 minutes.
Remove the foil and check to make sure the vegetables are tender by poking them with a paring knife. You can pause the recipe here, let it sit at room temp for a couple hours, and finish it after the turkey comes out of the oven. Sprinkle the shredded cheese over the top of the gratin and continue baking for an additional 15-20 minutes or until cheese is melted and lightly browned.
Let the gratin cool slightly before serving. Garnish with fresh thyme leaves if desired and season with salt to taste.