Heat oven to 350 degrees. Butter a shallow 1 1/2-quart baking dish, and set dish aside. In a large bowl, whisk together the eggs, cream, milk, salt, pepper, and nutmeg.
Immerse the bread strips in the egg mixture. Immediately transfer strips to the baking dish, and arrange randomly on the bottom. Pour the white wine and any remaining egg mixture over the bread strips. Sprinkle the grated cheese and thyme leaves over top.
Transfer baking dish to oven. Bake until the gratin is puffy, the center has set, and the cheese is brown and bubbly, 30 to 40 minutes. Remove from oven, and serve.