Preheat oven to 350°F. Coat a large baking sheet with cooking spray.
Cut off the neck of the squash and peel it. Slice into twelve 1/2-inch-thick rounds; remove any stray fibrous threads or seeds in the centers. (Reserve the rest of the squash for another use, such as squash puree). Place the squash rounds on the prepared baking sheet. Cover with foil.
Bake for 20 minutes. Remove the foil and continue baking until the squash is tender, about 20 minutes more.
Meanwhile, heat oil in a large skillet over medium heat. Add shallots, brown sugar and salt and reduce heat to medium-low; cook, stirring often, until the shallots are softened and beginning to brown, about 5 minutes. Stir in cider and vinegar and cook until most of the liquid is evaporated, 6 to 8 minutes.
To serve, place 1 squash round on a small plate and top with about 2 1/2 tablespoons shallot sauce. Place a second round on top and finish with another 2 1/2 tablespoons of sauce and a sprinkling of walnuts. Repeat with the remaining squash, shallot sauce and walnuts.
Look for a squash with a long slender neck like a goose¿at least 6 inches long and about 3 inches wide.To toast chopped walnuts, heat a small dry skillet over medium-low heat. Add nuts and cook, stirring, until lightly browned and fragrant, 2 to 3 minutes.