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In a small saucepan, bring the half-and-half to a simmer. Add the garlic, salt, pepper, thyme, oregano, and nutmeg, then set the pan aside to cool. Generously butter a 1-1/2-quart round casserole and preheat the oven to 375°F. Wash and peel the potatoes and beets. Slice them into 1/8-inch-thick rounds. Arrange the potatoes in one layer, in the bottom of the casserole. Combine the cheeses and sprinkle 1/3 cup over the potatoes. Add a layer of beets, overlapping as needed, and sprinkle with another 1⁄3 cup of cheese. Alternate the layers of potatoes and beets, with 1⁄3 cup of cheese between each, until they reach the rim of the casserole. Press the layers down to compact them and top with the remaining cheese. When the half-and-half mixture has reached room temperature, add the egg, whisking vigorously, and pour the mixture over the potatoes and beets. Cover the casserole loosely with foil and set it on a large baking sheet. Bake for 1 hour, or until the vegetables feel tender when pierced. Remove the foil and turn the oven to broil. Broil for 3 to 5 minutes, or until the cheese is lightly browned. Remove from the oven and let stand for 15 to 20 minutes. Cut into wedges and serve.