Bring 2 1/2 cups cider to a boil in a small saucepan over medium-high heat. Cook until reduced to 1/2 cup (about 20 minutes); cool. Cover and chill.
Preheat oven to 400°.
Cut squash in half lengthwise; discard seeds and membrane. Place squash halves, cut sides down, on a baking sheet; bake at 400° for 1 hour or until tender. Cool. Remove pulp; set aside.
Place grated ginger on several layers of paper towels. Gather the edges of paper towels together; squeeze paper towel bag over a small bowl, reserving ginger juice. Discard solids. Set aside.
Combine 1 1/2 cups cider, water, leek, and salt in a Dutch oven. Bring to a boil. Cover, reduce heat, and simmer 40 minutes or until leek is soft.
Add squash, ginger juice, and milk to leek mixture, stirring well. Cook 15 minutes. Place half of squash mixture in a blender or food processor; process until smooth. Pour pureéd squash mixture into a large bowl. Repeat procedure with remaining squash mixture. Return pureéd squash mixture to pan. Keep warm. Ladle soup into bowls; drizzle with cider syrup.