Adjust oven rack to middle position and preheat oven to 450°F (230°C). Grease bottom and sides of a 17 ½-by-12 ½-inch rimmed baking sheet with 1 tablespoon of the butter. In a large skillet, melt remaining 2 tablespoons butter over medium heat. Add onion and ½ teaspoon salt and cook, stirring occasionally, until onion is softened and starts to turn golden brown, about 10 minutes. Add garlic and cook, stirring constantly, until fragrant, about 1 minute. Add sherry and cook, stirring constantly, until evaporated, about 1 minute. Reduce heat to low, and add cream, thyme, sage, pepper, nutmeg, and ½ kosher teaspoon salt; bring to a simmer over low heat, then immediately remove from heat.

While cream mixture sits, thinly slice parsnips, sweet potatoes, and beets on a mandoline into ⅛-inch-thick rounds. Arrange half of the vegetable slices in an attractive pattern on prepared baking sheet so that bottom of sheet is not showing, tightly shingling or overlapping as needed. Sprinkle evenly with 1 teaspoon kosher salt and ⅔ cup Parmigiano-Reggiano. Top with half of remaining vegetable slices, tightly shingling or overlapping as needed, and sprinkle with remaining 1 teaspoon kosher salt. Using a ladle, pour warm cream mixture evenly over vegetables.

Cover baking sheet tightly with aluminum foil; bake until vegetable slices are just tender when pierced with a fork, 20 to 30 minutes. Carefully remove foil, and sprinkle evenly with Gruyère and remaining ⅔ cup Parmigiano-Reggiano.

Return uncovered baking sheet to oven; bake at 450°F, until browned and crusty on top, rotating baking sheet halfway through, about 20 minutes total.

Remove from oven, and let sit for 5 minutes. Sprinkle with additional thyme leaves, sage leaves, and black pepper. Serve immediately.