▢2 medium sweet potato, peeled, sliced into ½” rounds ▢2 medium carrots, peeled, sliced into ½” rounds ▢1 medium parsnip, peeled, sliced into ½” rounds ▢¼ cup olive oil ▢2 tablespoons thyme leaves and extra for serving ▢½ teaspoon nutmeg ▢⅛ teaspoon freshly ground black pepper ▢½ teaspoon salt ▢¼ cup sugar ▢¼ cup water ▢1 16-ounce frozen puff pastry, thawed ▢2 ounces Gruyere, thinly sliced ▢1 large egg, lightly beaten