Preheat oven to 425°F. Combine vegetables in a large bowl. Combine oil, 2 tablespoons butter, and anchovies in a small skillet and cook over medium heat, stirring and mashing anchovies with a wooden spoon until they dissolve, about 2 minutes. Add garlic and stir until garlic is softened but not browned, about 1 minute. Stir in thyme, salt, pepper, and nutmeg. Pour mixture over vegetables; toss gently to coat. Butter baking dish. Arrange vegetables upright in dish, alternating the vegetables and placing them in tight concentric circles. Bake until vegetables are almost tender and lightly golden, about 20 minutes. Remove from oven and sprinkle with cheese. Bake until cheese is melted and vegetables are fully cooked, 10 minutes more.