Heat oven to 450 degrees. Place garlic in an ovenproof ramekin; drizzle with olive oil. Roast until light brown and very soft, about 20 minutes. Remove garlic from oven, and let stand until cool enough to handle. Peel garlic, cut into slices, and set aside. Reduce heat to 425 degrees.
Meanwhile, peel potatoes, and slice them as thinly as possible; place them in a bowl, and put damp paper towel on top to keep them from turning brown. Peel rutabaga, and cut in half, slice as thinly as possible, and cover with a damp paper towel.
In a 10-inch nonstick ovenproof skillet, melt 2 tablespoons butter, swirling pan to coat bottom and sides. Remove from heat. Starting at the sides of the pan, arrange about half of the rutabaga slices in overlapping concentric circles, covering bottom of pan; press to compress. Sprinkle rutabaga with 1/2 teaspoon salt, 1/8 teaspoon pepper, 1/2 teaspoon thyme, and a third of the roasted garlic; dot with 1 tablespoon butter.
Arrange the potato slices in tight concentric circles over the rutabaga, and press down. Season with 1/2 teaspoon salt, 1/8 teaspoon pepper, 1/2 teaspoon thyme, and another third of the garlic; dot with 1 tablespoon butter. Arrange remaining rutabaga on top, and season again with 1/2 teaspoon salt, 1/8 teaspoon pepper, and remaining garlic; dot with 1 tablespoon butter.
Spread a large piece of aluminum foil with remaining tablespoon butter. Cover skillet tightly with the foil, buttered side down. Place a cast-iron skillet on the foil to weigh it down, and transfer to the oven. Bake until vegetables are tender when pierced with the tip of a knife, 50 to 60 minutes.
Remove pan from oven, and let stand on a wire rack for 15 minutes. Remove foil, and invert roots carefully onto a serving dish. Garnish with remaining 1/2 teaspoon thyme, and serve warm. This can be made ahead of time and reheated.